Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients... Department of Mathematics, Segregated direct boundary-domain integral equations (BDIEs) based on a Brunel University London, Uxbridge, UK parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE s–2 s–2 right-hand sides belong to the Sobolev (Bessel potential) space H ()or H (), 1 3 < s < , when neither strong classical nor weak canonical co-normal derivatives are 2 2 well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators. MSC: 35J25; 31B10; 45K05; 45A05 Keywords: Partial differential equations; Non-smooth coefficients; Sobolev spaces; Parametrix; Integral equations; Equivalence; Lipschitz domain; Invertibility 1 Introduction Many applications in science and engineering can be modelled by boundary-value prob- lems (BVPs) for partial differential equations with variable coefficients. Reduction of the BVPs with arbitrarily variable coefficients to explicit boundary integral equations is usu- ally not possible, since the fundamental solution needed for such reduction is generally not available in an analytical form (except for some special dependence of the coefficients on coordinates). Using a parametrix (Levi function) introduced in [20, 25]asasubsti- tute of a fundamental solution, it is possible however to reduce such a BVP to a system of boundary-domain integral equations, BDIEs, (see e.g. [38, Sect. 18], [43, 44], where the Dirichlet, Neumann, and Robin problems for some PDEs were reduced to indirect BDIEs). However, many questions about their equivalence to the original BVP, solvability, solution uniqueness, and invertibility of corresponding integral operators remained open for rather long time. In [3, 5, 6, 8, 30], the 3D mixed (Dirichlet–Neumann) boundary value problem (BVP) for the stationary diffusion PDE with infinitely smooth variable coefficient on a domain with an © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Mikhailov Boundary Value Problems (2018) 2018:87 Page 2 of 52 infinitely smooth boundary and a square-integrable right-hand side was reduced to either segregated or united direct boundary-domain integral or integro-differential equations, some of which coincide with those formulated in [29]. Such BVPs appear, for example, in electrostatics, stationary heat transfer, and other diffusion problems for inhomogeneous media. s 1 3 For a function from the Sobolev space H (), < s < , a classical co-normal derivative 2 2 in the sense of traces may not exist. However, the generalised co-normal derivative can be defined in the weak sense, associated with the first Green identity and with an extension s–2 of the corresponding second-order PDE right-hand side to H ()(see [27, Lemma 4.3], [31, Definition 3.1]). Since the extension is non-unique, the co-normal derivative operator appears to be also non-unique and non-linear in u unless a linear relation between u and the PDE right-hand side extension is enforced. This creates some difficulties in formulat- ing the boundary-domain integral equations. These difficulties are addressed in this paper presenting formulation and analysis of di- rect segregated BDIE systems equivalent to the Dirichlet and Neumann boundary value problems, on Lipschitz domains, for the divergent-type PDE with a non-smooth Hölder– s–2 Lipschitz variable scalar coefficient and a general right-hand side from H (), extended s–2 when necessary to H (). This needed a non-trivial generalisation of the third Green identity and its co-normal derivative for such functions, which essentially extends the ap- proach implemented in [3, 5, 6, 8, 30] for the right-hand side from L (), with smooth coefficient and smooth domain boundary. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm properties and invertibility of the BDIE operators are analysed in the Sobolev (Bessel potential) spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators. Some preliminary results in this direction for the infinitely smooth coefficient and domains were presented in [33]. Note that our analysis is mainly aimed not at the boundary-value problems, the proper- ties of which are well known nowadays, but rather at the BDIE systems per se. The analysis is interesting not only in its own rights but is also to be used further on for analysis of con- vergence and stability of BDIE-based numerical methods for PDEs; see, for example, [16, 29, 34, 35, 46–48, 52, 53]. 2 Spaces, co-normal derivatives and boundary value problems n n Let  =  be a bounded open n-dimensional region of R , n ≥ 3, and let  = R \ + – + denote the corresponding exterior domain. For simplicity, we assume that their common boundary ∂ is a simply connected closed Lipschitz surface. Let  denote  ,  or R . 0 + – ∞ n In what follows, D( ):= C ( )and D( ):= {r g : g ∈ D(R )}.Hereand fur- 0 0 0 comp 0 ther on, r denotes the restriction operator on  ; we will also use the equivalent s s s s notation g| := r g.Further, H ( )= H ( )and H (∂)= H (∂)are the Bessel 0 0 0 0 2 2 potential spaces, where s is a real number (see, e.g., [18, 26, 27]). We recall that H s s coincide with the Sobolev–Slobodetski spaces W for non-negative s.By H ( )we s n s n denote the closure of D( )in H (R ). It is a subspace of H (R ), and for Lipschitz s s n s s domains, H ( )= {g : g ∈ H (R ), supp g ⊂  }.By H ( )and H ( )we denote 0 0 0 0 s n s the spaces of restrictions on  of distributions from H (R )and H ( ), respec- 0 0 Mikhailov Boundary Value Problems (2018) 2018:87 Page 3 of 52 tively: s s n H ( ):= r g : g ∈ H R , s s s s H ( ):= r H ( ):= r g : g ∈ H ( ) ⊂ H ( ), 0  0  0 0 • 0 0 endowed by the corresponding infimum norms and the Hilbert structure defined with the s s help of orthogonal projections; see [27,p.77] for H ( ). Note that the space H ( ) 0 0 coincides with the one denoted as L ( )in [41, Eq. (5.2)] and [40, Eq. (2.212)] for s,z 0 p =2. s s n s n Let us introduce the subspace H := {g : g ∈ H (R ), supp g ⊂ ∂ } of H (R )(and of s s s H ( )). By H ( ) wedenotethe closureof D( )in H ( ). 0 0 0 0 Definition 2.1 Let E denote the operator of extension of functions g ∈ H ( ), s ≥ 0, to the whole R by zero outside  .By, e.g.,[27, Lemma 3.32 and Theorem 3.33] (see also s s 1 [31, Theorem 2.7]) the operator E : H ( ) → H ( ) is continuous if 0 ≤ s < ,and 0 0 0 2 1 1 1 we extend it also to the range – < s < defining it for – < s < 0 as (cf. the proof of [31, 2 2 2 Theorem 2.16]) s –s ˚ ˚ E g, v := g, E v , ∀ g ∈ H (), ∀ v ∈ H (). (2.1) 0 0 0 0 Remark 2.2 Note the following known or easily deduced results: s s s 1. There hold the continuous embeddings H ( ) → H ( ) → H ( );see [42, 0 0 0 Eq. (2.123)]. s s 1 2. H ( )= H ( ) for any s >1/2 such that s – is non-integer; see, e.g., [27, 0 0 Theorem 3.3]. s s 3. H ( )= H ( ) for any s ≤ 1/2;see [31, Theorem 2.12]. 0 0 s s s 1 4. H ( )= H ( )= H ( ) for any s <1/2 such that s – is non-integer; see, e.g., 0 0 0 [31, Lemma 2.15]. s s 5. For any s ∈ R, there evidently exists an extension from H ( ) to H ( ),and for 0 0 any s ≥ –1/2,thisextension is unique;see,e.g., [27, Lemma 3.39], [31, Theorem 2.10(i)]. 6. By [31, Theorem 2.16], for any s ∈ (–1/2, 1/2),the extensionfrom s s s s ˚  ˚ H ( )= H ( )= H ( ) to H ( ) is unique and is given by the operator E . 0 0 0 0 • 0 Remark 2.3 Due to Remark 2.2(5), for s ≥ –1/2, the space H ( ) is isometrically iso- morphic to the space H ( ), and sometimes these spaces are identified. Particularly, s s if g , g ∈ H ( ), then denoting by g ˜ , g ˜ ∈ H ( ) the unique distributions such that 1 2 0 1 2 0 g = r g ˜ in  ,wehave g s = g ˜ s and (g , g ) s =(g ˜ , g ˜ ) s .Moreover,if i  i 0 i H ( ) i H ( ) 1 2 H ( ) 1 2 H ( ) • 0 0 • 0 0 ˚ ˚ s ∈ (–1/2, 1/2), then by Remark 2.2(6), g ˜ = E g hence implying g s = E g . i  i i  i 0 H ( ) 0 H ( ) • 0 0 There is no such isomorphism for s < –1/2 since in such a case the extension from s s H ( )to H ( ) is not unique. However, due to the definition of the spaces, there is still 0 0 s s s an isometric isomorphism between the space H ( ) and the quotient space H ( )/H . 0 0 • ∂ Definition of the space H ( ), Remark 2.2,and Remark 2.3 imply the following asser- tion. Mikhailov Boundary Value Problems (2018) 2018:87 Page 4 of 52 Corollary 2.4 The following restriction operators are isomorphisms: s s r : H ( ) → H ( ), – ≤ s, (2.2) 0 0 0 • 1 1 s s s r : H ( ) → H ( )= H ( ), – < s < , (2.3) 0 0 0 0 • 2 2 s s s r : H ( )/H → H ( ), s <– . (2.4) 0 0 0 ∂ • –1 The inverse to the operator (2.3) is r = E ; see Definition 2.1. m,θ Definition 2.5 For a non-negative integer m and 0 < θ ≤ 1, let C ( )denote the Hölder–Lipschitz space in the closed domain  . Similar to [32, Definition 3.1], g ∈ ( )for μ ≥ 0means that + 0 g ∈ L ( ) when μ =0; ∞ 0 μ–1,1 g ∈ C ( ) when μ is a positive integer; m,θ+ g ∈ C ( ) for some  >0 when μ = m + θ,where m is a non-negative integer, and 0< θ <1. Employing this definition, Theorem 7.2 from Sect. 7 can be reformulated as follows. |σ | n σ Theorem 2.6 Let  be an open set in R , σ ∈ R, v ∈ H ( ), and g ∈ C ( ). Then g 0 0 + 0 σ σ σ is a multiplier in H ( ), i.e., gv ∈ H ( ) for every v ∈ H ( ), and the corresponding 0 0 0 norm estimate holds. Let us denote ∂ := ∂ := ∂/∂x (j = 1,2,..., n), ∇ =(∂ , ∂ ,..., ∂ ). Let j x j 1 2 n 0< a ≤ a(x) ≤ a < ∞ for almost every x ∈  . (2.5) min max ± We consider the scalar elliptic differential equation, which can be written in the following strong form if u and a are sufficiently smooth: Au(x):= A(x, ∇)u(x):= ∇· a(x)∇u(x) = f (x), x ∈  , (2.6) where u is an unknown function and f is a given function in  . |s–1| For u ∈ H ( ), 1/2 < s <3/2, and a ∈ C ( ), the partial differential operator A is ± + ± understood in the sense of distributions: Au, v := –E (u, v), ∀v ∈ D( ), (2.7) ± ± where E (u, v):= a∇u, ∇v := a∂ u, ∂ v , i i ± ± ± i=1 and the duality brackets g, · denote value of a linear functional (distribution) g extend- ing the usual L dual product. If s =1, then E (u, v)= a(x)∇u(x) ·∇v(x) dx. ± Mikhailov Boundary Value Problems (2018) 2018:87 Page 5 of 52 2–s Since the set D( )is dense in H ( ), (2.7) defines, due to Theorem 2.6 (see, e.g., ± ± s s–2 2–s ∗ [32, Theorem 3.4]), the continuous linear operator A : H ( ) → H ( )=[H ( )] , ± ± ± where s 2–s Au, v := –E (u, v), ∀ u ∈ H ( ), v ∈ H ( ). (2.8) ± ± ± ± s s–2 2–s ∗ Let us also consider the operator A : H ( ) → H ( )=[H ( )] (see [31, ± ± ± Eq. (3.5)], [32, Eq. (5.1)]) defined by ˇ ˇ A u, v := –E (u, v):=– E (a∇u), ∇v ± ± ± ± ˚ ˚ =– E (a∇u), ∇v = ∇· E (a∇u), v e n  e n ± ± R R s 2–s = ∇· E (a∇u), v , ∀ u ∈ H ( ), v ∈ H ( ), (2.9) ± ± 2–s n which is evidently continuous. Here v ∈ H (R )is such that r v = v. Evidently, weak e  e definition (2.9) can be also written (in the strong-looking form) as A u = ∇· E r [a∇u]. (2.10) ± ± ± s s–2 For any u ∈ H ( ), the functional A u belongs to H ( ) and is a specific extension ±  ± s–2 s–2 of the functional Au ∈ H ( ); recall that the functional Au ∈ H ( )is defined on ± ± 2–s 2–s H ( ), whereas the functional A u is defined on H ( ). ±  ± Remark 2.7 Note also that Definition 2.1 for E and definition (2.9)imply that ˇ ˇ ˇ ˇ A u, v =–E (u, v)=–E (v, u)= u, A v , ± ± ± ± ± ± s 2–s ∀ u ∈ H ( ), v ∈ H ( ), 1/2 < s < 3/2. ± ± From the trace theorem (see, e.g., [11, 12, 26, 27]) for u ∈ H ( ), 1/2 < s < 3/2, it follows s– ± ± ± + – that γ u ∈ H (∂), where γ = γ is the trace operator on ∂ from  .If γ u = γ u, –1 –1 s– s n then we will sometimes write just γ u.Let also γ := γ : H (∂) → H (R )denote –1 a (non-unique) continuous right inverse to the trace operator γ , i.e., γγ w = w for any 1 1 s– ± –1 s– 2 2 w ∈ H (∂). Hence also γ γ w = w for any w ∈ H (∂). s c± For u ∈ H ( ), s > ,and a ∈ C( ), we denote by T the corresponding classical ± ± (strong) co-normal derivative operators on ∂ in the sense of traces: c± ± T u(x):= a(x)ν(x) · γ ∇u(x)= a(x)∂ u(x), x ∈ ∂, (2.11) where ν(x)= ν (x) is the outward to  unit normal vector at the point x ∈ ∂,and we will c c+ c– sometimes write T u(x)if T u(x)= T u(x). However, the classical co-normal derivative is, generally, not well defined if u ∈ H ( ), 1/2 < s < 3/2,(seeanexamplein[33,Ap- pendix A] of a function from H (), wherethe classicalnormalderivativedoesnot exist at boundary points). Inspired by the first Green identity for smooth functions, we can define the generalised co-normal derivative (cf., e.g., [27, Lemma 4.3]), [31, Definition 3.1], [32, Definition 5.2]). Mikhailov Boundary Value Problems (2018) 2018:87 Page 6 of 52 |s–1| Definition 2.8 Let 1/2 < s < 3/2, u ∈ H ( ), a ∈ C ( ), and r Au = r f for some ± + ±   ± ± ± s–2 ± s– ˜ ˜ f ∈ H ( ). Then the generalised co-normal derivatives T (f ; u) ∈ H (∂)are de- ± ± ± fined in the weak form as ± –1 –1 ˜ ˜ ˇ ± T (f ; u), w := f , γ w + E u, γ w ± ± –1 –s ˜ ˇ = f – A u, γ w , ∀ w ∈ H (∂), (2.12) i.e., ± –1 ˜ ˜ ˇ T (f , u):= ± γ (f – A u). (2.13) ± ± If a ≡ 1, then A = ,and T (f ; u)become generalised normal derivatives denoted as T (f ; u). 1 1 –1 ∗ –t n –t+ –1 t– t n 2 2 The operator (γ ) : H (R ) → H (∂)is dual to γ : H (∂) → H (R )and is –1 ∗ –1 t– –t n defined as (γ ) ψ, w := ψ, γ w n for any w ∈ H , ψ ∈ H (R ), 1/2 < t < 3/2. In ∂ R (2.13)itwas employed for t =2 – s. Theorem 2.9 (Lemma 4.3 in [27], Theorem 3.2 in [31], and Theorem 5.3 in [32]) Un- der the hypotheses of Definition 2.8, the generalised co-normal derivatives T u(f ; u) are –1 independent of (non-unique) choice of the operator γ , and we have the estimate ˜ ˜ T (f ; u) 3 ≤ C u s + C f s–2 (2.14) ± 1 H ( ) 2 ± H ( ) s– ± ± H (∂) and the first Green identity in the form ± ± ˜ ˜ ˇ ± T (f ; u), γ v = f , v + E (u, v) ± ± ± ± 2–s ˜ ˇ = f – A u, v , ∀ v ∈ H ( ). (2.15) ±   ± ± ± As follows from Definition 2.8, the generalized co-normal derivative is nonlinear with re- ˜ ˜ spect to u for fixed f but still linear with respect to the couple (f , u), i.e., for any complex ± ± numbers α and α , 1 2 ± + ± ± ˜ ˜ ˜ ˜ α T (f ; u )+ α T (f ; u )= T (α f ; α u )+ T (α f ; α u ) 1 1± 1 2 2± 2 1 1± 1 1 2 2± 2 2 ˜ ˜ = T (α f + α f ; α u + α u ). 1 1± 2 2± 1 1 2 2 Let us also define some subspaces of H ( ); see [11, 15, 31, 32]. s ∗ Definition 2.10 Let s ∈ R,and let A : H ( ) → D ( ) be a linear operator. For t ∈ R, ∗ ± ± we introduce the space s,t s t H ( ; A ):= g : g ∈ H ( ), A g ∈ H ( ) ± ∗ ± ∗ ± 2 2 1/2 s,t endowed with the norm g := ( g + A g ) and the correspond- H ( ;A ) s ∗ t ± ∗ H ( ) ± H ( ) • ± ing inner product. Mikhailov Boundary Value Problems (2018) 2018:87 Page 7 of 52 s,t Definition 2.11 Let  be either  or  .ByRemark 2.3,if g ∈ H ( ; A )for some 0 + – 0 ∗ 1 t ˜  ˜ s ∈ R and t ≥ – , then there exists a unique distribution f ∈ H ( )such that r f = A g, 0  ∗ 2 0 –1 s,t t ˜ ˜ ˜ ˜ and hence f = A g,where A := r A . The operator A : H ( ; A ) → H ( ), ∗ ∗ ∗ ∗ 0 ∗ 0 0 0  0 which is continuous by Corollary 2.4, is called the canonical extension of the operator s,t t 1 1 ˜ ˚ A : H ( ; A ) → H ( ), and moreover, if – < t < ,then A = E A . ∗ 0 ∗ 0 ∗  ∗ • 2 2 0 0 We will mostly use the operators A or as A in the definition. Note that since Au = 1 1 s,– s,– 2 2 2 u + ∇a ·∇u,for 1/2 < s < 3/2, we have H ( ; A)= H ( ; )if a ∈ C ( ), with 0 0 + 0 equivalent norms. Let us now define the canonical conormal derivative; see [32, Definition 6.5]. |s–1| s,– Definition 2.12 For u ∈ H ( ; A)and a ∈ C ( ), 1/2 < s < 3/2, we define the ± + ± s– canonical co-normal derivatives T u ∈ H (∂)as ± –1 –1 –1 ˜ ˜ ˇ ± T u, w := A u, γ w + E u, γ w = A u – A u, γ w ± ± ± ± ± ± –1 –s ˜ ˇ = γ (A u – A u), w ∀ w ∈ H (∂), (2.16) ± ± i.e, ± –1 ˜ ˇ T u := ± γ (A u – A u). (2.17) ± ± ± ± If a ≡ 1, then T u becomes the canonical normal derivative denoted as T u. Theorem 2.13 (Theorem 3.9 in [31]and Theorem6.6 in [32]) Under the hypotheses of Definition 2.12, the canonical co-normal derivatives T uare independent of (non-unique) 1 3 –1 ± s,– s– 2 2 choice of the operator γ , the operators T : H ( ; A) → H (∂) are continuous, and the first Green identity holds in the form ± ± ˜ ˇ ± T u, γ v = A u, v + E (u, v) ± ± ± 2–s ˜ ˇ = A u – A u, v , ∀ v ∈ H ( ). (2.18) ± ± ± The canonical co-normal derivatives in Definition 2.12 are completely defined by the func- tion u and operator A only and do not depend explicitly on the right-hand sides f ,un- like the generalised co-normal derivatives defined in (2.15), whereas the operators T are linear in u. Note that the canonical co-normal derivatives coincide with the classical co- ± c± normal derivatives T u = T u if the latter do exist (see [32, Corollaries 6.11 and 6.14]), which is generally not the case for the generalised conormal derivatives even for smooth ˜ ˜ functions u,unless f = A u is chosen. Thus the canonical conormal derivative is a con- tinuous extension of the classical conormal derivative. |s–1| s,– Let 1/2 < s < 3/2 and a ∈ C ( ). If u ∈ H ( ; A), then Definitions 2.8 and 2.12 + ± ± s–2 imply that the generalised co-normal derivative for arbitrary extensions f ∈ H ( )of ± ± the distributions r Au can be expressed as ± ± –1 ˜ ˜ ˜ T (f ; u)= T u ± γ (f – A u). (2.19) ± ± ± Mikhailov Boundary Value Problems (2018) 2018:87 Page 8 of 52 s 2–s,– If u ∈ H ( )and v ∈ H ( ; A), then swapping over the roles of u and v in (2.18), ± ± we obtain the first Green identity for v: ± ± ˇ ˜ ± T v, γ u = E (v, u)+ A v, u . (2.20) ± ± ± s–2 ˜ ˜ If, in addition, r Au = r f ,where f ∈ H ( ), then subtracting (2.20)from(2.15) ± ± ± ± ± ˇ ˇ and taking into account that E (u, v)= E (v, u)by Remark 2.7, we obtain the following ± ± second Green identity: ± ± ± ± ˜ ˜ ˜ f , v – A v, u = ± T (f ; u), γ v ∓ T v, γ u . ±    ± ± ± ± ∂ ∂ 1 1 s,– 2–s,– 2 2 If, finally, u ∈ H ( ; A)and v ∈ H ( ; A), then we arrive at the familiar form of ± ± the second Green identity for the canonical extension A of the operator A and the canon- ical co-normal derivatives ± ± ± ± ˜ ˜ A u, v – A v, u = ± T u, γ v ∓ T v, γ u . (2.21) ± ± ± ± ∂ ∂ 3 Parametrix and potential type operators on Lipschitz domains Recall that unless stated otherwise, we will assume that  =  . We will say that a function P(x, y)of two variables x, y ∈ R is a parametrix (the Levi function) for the operator A(x, ∂ )in R if (see, e.g., [19, 20, 25, 29, 38, 43, 44]) A(x, ∇ )P(x, y)= δ(x – y)+ R(x, y), (3.1) where δ(·) is the Dirac distribution, and R(x, y) possesses a weak (integrable) singularity at x = y, i.e., –κ R(x, y)= O |x – y| with κ < n. (3.2) n/2 2π n Let ω = denote the area of the unit sphere in R . It is well known that the function (n/2) –1 P (x, y)= , x, y ∈ R , (3.3) n–2 (n –2)ω |x – y| is the fundamental solution of the Laplace equation, i.e., P (x, y)= P (x, y)= δ(x – y). It is easy to see that for the operator A(x, ∂ ) given by the left-hand side in (2.6), the function 1 –1 P(x, y)= P (x, y)= , x, y ∈ R , (3.4) n–2 a(y) (n –2)ω a(y)|x – y| is a parametrix, whereas the corresponding remainder function is R(x, y)= ∇a(x) ·∇ P(x, y)= ∇a(x) ·∇ P (x, y) x x a(y) (x – y) ·∇a(x) = , x, y ∈ R , (3.5) ω a(y)|x – y| n Mikhailov Boundary Value Problems (2018) 2018:87 Page 9 of 52 1 n and if a ∈ C (R ), then it satisfies estimate (3.2)a.e.with κ = n –1. Note also that A(y, ∇ )P(x, y)= δ(x – y)+ R (x, y), (3.6) y ∗ where R (x, y)=–∇ · P(x, y)∇a(y) ∗ y (ln a(y)) (x – y) ·∇a(y) = – , x, y ∈ R . (3.7) n–2 n (n –2)ω |x – y| ω a(y)|x – y| n n Evidently, the parametrix P(x, y)given by (3.4) is related to the fundamental solution to the operator A(y, ∇ ):= a(y) with “frozen” coefficient a(y), and A(y, ∇ )P(x, y)= δ(x – y). x x x Note that parametrix (3.4) and remainders (3.5)and (3.7) are not smooth enough for the corresponding potential operators to be directly treated as in [27], which thus need some additional consideration. For g ∈ D(R ) and sufficiently smooth coefficient a, the parametrix-based volume po- tential operator and the remainder potential operator corresponding to parametrix (3.4) and remainders (3.5)and (3.7)for y ∈ R are Pg(y):= P(·, y), g = P(x, y)g(x) dx, (3.8) Rg(y):= R(·, y), g = R(x, y)g(x) dx, (3.9) R g(y):= R (·, y), g = R (x, y)g(x) dx, (3.10) ∗ ∗ ∗ and from (3.1)–(3.10)weobtain, PAg = g + Rg, APg = g + R g in R . (3.11) For the function g defined on a domain  ⊂ R , e.g., g ∈ D( ), the corresponding + + potentials for y ∈  are Pg(y):= P(·, y), g = P(x, y)g(x) dx, (3.12) Rg(y):= R(·, y), g = R(x, y)g(x) dx, (3.13) R g(y):= R (·, y), g = R (x, y)g(x) dx. (3.14) ∗ ∗ ∗ From definitions (3.4), (3.5), and (3.7) we can obtain representations of the parametrix- based potential operators in terms of their counterparts for a = 1 (i.e., associated with the Laplace operator ; see, e.g., [21]), which we equip with the subscript (see [3]): 1 1 ∇a Pg = P g, Rg =– ∇· P (g∇a), R g =–∇· P g , (3.15) a a a 1 1 ∇a Pg = P g, Rg =– ∇· P (g∇a), R g =–∇· P g . (3.16) a a a Mikhailov Boundary Value Problems (2018) 2018:87 Page 10 of 52 Hence (aPg)= g in R , (aPg)= g in . (3.17) Employing relations (3.16) and the well-known properties of the operator P as the pseudo-differential operator of order –2 together with Theorem 2.6, definitions (3.8)– s n s (3.10) can be extended to g ∈ H (R ), g ∈ H () and lower-smoothness coefficient a.For s s g ∈ H ()and g ∈ H (), the potentials P, R, R defined on functions (or distributions) having support on  are understood as Pg := r Pg, Rg := r Rg, R g := r R g, g ∈ H (), s ∈ R; (3.18) ∗  ∗ ˚ ˚ ˚ Pg := r PE g, Rg := r RE g, R g := r R E g, g ∈ H (), s >– . (3.19) ∗  ∗ To prove mapping properties of the parametrix-based volume potential operators in Sobolev spaces, we first provide some well-known results for the classical Newtonian vol- ume potential associated with the Laplace operator. Lemma 3.1 Let  be a bounded Lipschitz domain in R . The following operators are con- tinuous: s–2 n s n n μP : H R → H R , s ∈ R, ∀ μ ∈ D R ; (3.20) s–2 s P : H () → H (), s ∈ R; (3.21) 3 5 s–2 s P : H () → H (), < s < ; (3.22) 2 2 1 3 s–2 s,– P : H () → H (; ), s ≥ ; (3.23) 1 1 3 + s–2 s– γ P : H () → H (∂), < s < ; (3.24) 2 2 + s 1 γ P : H () → H (∂), – < s; (3.25) + s T P : H () → L (∂), – < s; (3.26) + s T P : H () → L (∂), – < s. (3.27) 1 3 s–2 s–2 ˜  ˜  ˜ ˜ If < s < , f ∈ H (), and f ∈ H () is such that r f = r f , then there exist constants 0  0 2 2 C , C >0 such that 0 1 ˜ ˜ ˜ ˜ T (f ; P f ) 3 ≤ C f s–2 + C f s–2 . (3.28) 1  2 0 s– H () H () H (∂) Proof Operator (3.20) and hence (3.21)are continuous since P is a pseudo-differential operator of order –2. The continuity of operator (3.22) follows from the first relation in (3.19)for P and P and from (3.21). Since P g = g in , the continuity of operator (3.21) implies that of operator (3.23). Mikhailov Boundary Value Problems (2018) 2018:87 Page 11 of 52 The continuity of operator (3.24) is implied by that of operator (3.21) and the trace the- orem for Lipschitz domains; see, e.g., [11, Lemma 3.6] and [27, Theorem 3.38]. The con- tinuity of operator (3.25) follows from that of operator (3.22) and from, e.g., [54], [31, 1 1 1 Lemma 2.5] for – < s < , and then by the embedding argument for s ≥ . 2 2 2 The continuity of operators (3.26)and (3.27)isimplied by that of (3.21)and (3.22), re- spectively, and by [31, Corollary 3.14] since s +2 > in both cases. Estimate (3.28) follows ˜ ˜ from the continuity of operator (3.21), relation P f = f , and estimate (2.14). Now the following mapping properties of the parametrix-based operators can be ob- tained. Theorem 3.2 Let  be a bounded Lipschitz domain in R . The following operators are continuous: s–2 n s n |s| n n μP : H R → H R , s ∈ R, a ∈ C R , ∀ μ ∈ D R ; (3.29) s–2 s |s| P : H () → H (), s ∈ R, a ∈ C (); (3.30) 3 5 s–2 s s P : H () → H (), < s < , a ∈ C (); (3.31) 2 2 1 3 s–2 s,– s P : H () → H (; A), ≤ s, a ∈ C (); (3.32) s–1 n s n |s–1|+1 n n μR : H R → H R , s ∈ R, a ∈ C R , ∀ μ ∈ D R ; (3.33) 1 3 s–1 s |s–1|+1 R : H () → H (), < s < , a ∈ C (); (3.34) 2 2 1 3 s s s R : H () → H (), < s < , a ∈ C (); (3.35) 2 2 1 1 3 s s,– R : H () → H (; A), < s < , a ∈ C (); (3.36) 2 2 s n s+1 n |s+2|+1 n n μR : H R → H R , s ∈ R, a ∈ C R , ∀ μ ∈ D R ; (3.37) s s+1 |s+2|+1 R : H () → H (), s ∈ R, a ∈ C (); (3.38) 3 1 s σ 2 R : H () → H (), – < s, a ∈ C (), for some σ >– ; (3.39) ∗ + 2 2 1 1 3 + s–2 s– s γ P : H () → H (∂), < s < , a ∈ C (); (3.40) 2 2 + s 1 2 γ P : H () → H (∂), – < s, a ∈ C (); (3.41) 1 1 3 + s s– s γ R : H () → H (∂), < s < , a ∈ C (); (3.42) 2 2 + s 2 T P : H () → L (∂), – < s, a ∈ C (); (3.43) 2 + + s 2 T P : H () → L (∂), – < s, a ∈ C (); (3.44) 2 + 3 1 3 + s s– 2 T R : H () → H (∂), < s < , a ∈ C (). (3.45) 2 2 Moreover, operators (3.35), (3.36), (3.42), and (3.45) are compact. Mikhailov Boundary Value Problems (2018) 2018:87 Page 12 of 52 1 3 s s–2 s–2 ˜ ˜ ˜ ˜ If < s < , a ∈ C (), f ∈ H (), and f ∈ H () is such that r f = r APf , then 0  0 2 2 there exist constants C , C >0 such that 0 1 ˜ ˜ ˜ ˜ T (f ; Pf ) 3 ≤ C f s–2 + C f s–2 . (3.46) 0 1  2 0 s– H () H () H (∂) Proof The continuity of operators (3.29)–(3.31) is implied by the first relations in (3.15) and (3.16) and by the continuity of operators (3.20)–(3.22) together with Theorem 2.6. The continuity of operators (3.30)and (3.31)and Remark 2.2(4) imply that of operator 1 3 3 3 2 2 (3.32)for s > .Let us nowprove (3.32)for s = .For g ∈ H (), we have, Pg ∈ H () 2 2 due to (3.30), whereas APg = ∇· a∇ P g = g – ∇· (∇ ln a)P g in , (3.47) wherewehavetaken into account that P g = g. The first term in the right-hand side of 2 σ σ (3.47)belongs to H (), whereas the second term belongs to H ()= H ()for some σ ∈ (–1/2, 1/2) (cf. item 4 in Remark 2.2)since ∇a ∈ C ()and a ≥ a >0, which com- + min pletes the proof of the continuity of operator (3.32). The continuity of operator (3.33) follows from the second relation in (3.15)togetherwith Theorem 2.6 and the continuity of operator (3.20). Indeed, let us take arbitrary μ ∈ D(R ), let B be a ball such that supp μ ⊂ B ,and let μ ∈ D(R )be such that μ =1 in B .Then μ μ 1 1 μ s–1 n for any g ∈ H (R ), we have μRg s n = ∇· μ P (g∇a) ≤ c ∇· μ P (g∇a) H (R ) 1 1 1 s n H (R ) s n H (R ) ≤ c μ P (g∇a) ≤ c g∇a s–1 n ≤ c g s–1 n , (3.48) 2 1 3 4 s+1 n H (R ) H (R ) H (R ) where c are positive constants (depending on μ, μ ,and a), and we took into account that i 1 |s–1|+1 |s| n n C (R ) ⊂ C (R )since |s|≤|s –1| +1. + + To prove the continuity of operator (3.34), we similarly employ the second relation in (3.16) together with Theorem 2.6 and the continuity of operator (3.22). Then we obtain s–1 for any g ∈ H (), 1/2 < s < 3/2, and some positive constants c : Rg = ∇· P (g∇a) ≤ c ∇· P (g∇a) H () H () H () ≤ c P (g∇a) ≤ c g∇a s–1 ≤ c g s–1 . (3.49) 3 4 s+1 H () H () H () Let us prove the continuity and compactness of operator (3.35). For 1 ≤ s < ,wehave s = |s –1| + 1, and then the continuity of operator (3.34) implies the continuity and com- pactness of (3.35). For < s < 1, we need a sharper estimate of the norm g∇a s–1 . H () First, by Definition 2.5 the inclusion a ∈ C () implies that there exists t ∈ (s,1) such 0,t t t that a ∈ C ()= B ()= F (); see, e.g., Proposition in [45, Sect. 2.1.2], and hence ∞,∞ ∞,∞ t–1 ∇a ∈ F (). Then, by Theorems 1 from [45, Sect. 4.4.3] we have ∞,∞ g∇a ≤ C ∇a g σ t–1 t–1 H () F () F () ∞,∞ 2,∞ ≤ C a g σ , ∀ σ ∈ (1 – t, s). (3.50) 0,t H () C () Mikhailov Boundary Value Problems (2018) 2018:87 Page 13 of 52 On the other hand, by (3.49), item (ii) of Proposition from [45, Sect. 2.2.1], and (3.50)we obtain Rg s ≤ c g∇a = c g∇a s–1 s–1 H () 3 3 H () F () 2,2 ≤ C g∇a t–1 ≤ C C a g σ . 0,t 1 1 H () F () C () 2,∞ σ s Thus the operator R : H () → H () is continuous, which implies the continuity and, by the Rellich compact embedding theorem, also the compactness of operator (3.35)for < s <1. Let us prove the continuity of operator (3.36). Since a ∈ C (), by Definition 2.5 there 1, + 1 1 exists  >0 such that a ∈ C (), and let us choose any σ ∈ ( , min{s, + }). By the 2 2 σ s continuity of (3.34) the operator R : H () → H () is continuous. Now let us prove σ 2 that the operator AR : H () → H () is continuous as well. Indeed, for some positive constants c ,wehave ARg 1 ≤ ARg σ –1 – H () H () ≤ c ARg σ –1 0 H () = c ∇· a∇ ∇· P (g∇a) σ –1 H () = c –∇· (∇ ln a)∇· P (g∇a) + ∇· P (g∇a) σ –1 H () ≤ c –(∇ ln a)∇· P (g∇a)+(g∇a) H () ≤ c a 1 P (g∇a) + c g∇a 1 H () σ +1 1, + H () σ σ ≤ c g∇a σ –1 + c g∇a ≤ c g∇a 3 1 H () 4 H () H () ≤ c a g σ . 5 H () 1, + C 2 σ s,– Hence we proved the continuity of the operator H () → H (; A), which implies that of operator (3.36) and by the Rellich compact embedding theorem also its compactness. The continuity of operator (3.37)isimplied by thelastrelationin(3.15), the continuity of operator (3.20), and Theorem 2.6 in the chain of inequalities analogous to (3.48). Similarly, the continuity of operator (3.38)isimplied by thelastrelationin(3.16), the continuity of operator (3.21), and Theorem 2.6. The continuity of operator (3.39)isimplied by that of 2 1,1/2+ (3.38)since a ∈ C () implies that there exists  >0 such that a ∈ C (), and we can 3 3 take σ ∈ ( , min{s +1, + }). 2 2 The continuity of operator (3.40) is implied by that of operator (3.30) and the trace theo- rem for Lipschitz domains; see, e.g., [11, Lemma 3.6], [27, Theorem 3.38]. The continuity 1 1 of operator (3.41)for – < s <– +  with any sufficiently small  > 0 follows from that 2 2 of operator (3.31) together with, e.g., [54], [31, Lemma 2.5] and then by the embedding argument for all s >– . Similarly, the continuity of operators (3.43)and (3.44)isimplied by that of (3.30)and (3.31), respectively, and by [31, Corollary 3.14] since s +2 > in the both cases. Mikhailov Boundary Value Problems (2018) 2018:87 Page 14 of 52 The continuity and compactness of operators (3.42)and (3.45) are implied by those of operators (3.35)and (3.36), the trace theorem for Lipschitz domains, and Theorem 2.9. Estimate (3.46) follows from the continuity of operator (3.30) and estimate (2.14). The parametrix-based single- and double-layer surface potential operators are defined as Vg(y):=– P(x, y)ψ(x) dS , y ∈/ ∂, (3.51) Wg(y):=– T x, n(x), ∂ P(x, y) ϕ(x) dS , y ∈/ ∂, (3.52) x x where the integrals are understood as duality forms if ψ and ϕ are not integrable. Partic- 1 1 1 3 –s –s 2 2 ularly, for ψ ∈ H (∂)and ϕ ∈ H (∂), < s < ,wehave 2 2 V ψ(y):= – γ P(·, y), ψ =– P(·, y), γ ψ ∂ R ∗ ∗ =– Pγ ψ(y)=– P γ ψ(y), (3.53) a(y) c c∗ W ϕ(y):= – T P(·, y), ϕ =– P(·, y), T ϕ ∂ R c∗ c∗ =– PT ϕ(y)=– P T ϕ(y), (3.54) a(y) ∗ c∗ –1 where γ ψ and T ϕ are well defined for any ψ ∈ H (∂), ϕ ∈ L (∂), and a ∈ L (∂), 2 ∞ in the sense of distributions, as ∗ n γ ψ, φ := ψ, γφ , ∀ φ ∈ D R ,and n ∂ c∗ c c n T ϕ, φ := ϕ, T φ = ϕ, aT φ , ∀ φ ∈ D R , R ∂ ∂ ∗ c∗ which evidently implies that supp γ ψ ⊂ ∂ and supp T ϕ ⊂ ∂.Moreover, 1 1 1 3 ∗ –s –s c∗ –s –s–1 2 2 γ : H (∂) → H , T : H (∂) → H , < s < , (3.55) ∂ ∂ 2 2 are the continuous operators adjoint, respectively, to the continuous trace operator γ : s n s– c H (R ) → H (∂) and to the continuous classical conormal derivative operator T : loc s+1 n s– c c∗ H (R ) → H (∂); for the continuity of T and T ,itisalsoassumed that a ∈ loc s– C (∂). When a =1, formulas (3.51)and (3.52) define the corresponding harmonic potentials, which we denote as V and W , respectively. From definitions (3.51)and (3.52), similar to (3.15)–(3.16), we have (cf. [3]) 1 1 Vg = V g, Wg = W (ag). (3.56) a a Hence (aVg)=0, (aWg)=0 in  . (3.57) ± Mikhailov Boundary Value Problems (2018) 2018:87 Page 15 of 52 We will mainly need the restrictions of the layer potentials to , i.e., r V and r W,but will often omit the restriction operator r if this is clear from the context. The mapping properties and jump relations for the single- and double-layer potentials are well known for the case a = const and were extended to the case of infinitely smooth boundary and variable coefficient a(x)in [3, 5]. Before proving the corresponding prop- erties for the parametrix-based potentials on Lipschitz domains, we further collect the following well-known mapping and jump properties for the harmonic potentials on Lips- chitz domains. Theorem 3.3 Let  be a bounded Lipschitz domain in R . 1 3 n (i) If ≤ s ≤ , then the following operators are continuous for any μ ∈ D(R ): 2 2 s– s n μV : H (∂) → H R , (3.58) 1 1 s– s s– s 2 2 r W : H (∂) → H (), μr W : H (∂) → H ( ). (3.59) 1 3 (ii) If < s < , then the following operators are continuous: 2 2 3 1 1 1 ± s– s– ± s– s– 2 2 2 2 γ V : H (∂) → H (∂), γ W : H (∂) → H (∂), (3.60) 3 3 1 3 ± ± s– s– s– s– 2 2 2 2 T V : H (∂) → H (∂), T W : H (∂) → H (∂), (3.61) 1 3 1 3 s– s– 2 2 (iii) If < s < , then, for any ϕ ∈ H (∂) and ψ ∈ H (∂), the following jump 2 2 properties hold: + – + – γ V ψ – γ V ψ =0, γ W ϕ – γ W ϕ =–ϕ, (3.62) + – + – T V ψ – T V ψ = ψ, T W ϕ – T W ϕ = 0. (3.63) Proof Items (i) and (ii) follow, e.g., from [11, Theorem 1(i,ii) and Remark], [22–24, 51](see also [27, Theorem 6.12]) if we take into account that the canonical co-normal derivative operators in (3.61)are well defined since V =0and W =0in  .The jump properties of item (iii) for s = 1 are implied, e.g., by [11, Lemma 4.1]; see also [27, Theorem 6.11]. 3 1 Hence they evidently hold if 1 ≤ s < and by the density argument also if < s <1. 2 2 Theorem 3.3 implies the following assertion. 1 3 Corollary 3.4 Let ∂ be a compact Lipschitz boundary, and let < s < . The following 2 2 operators are continuous: 3 1 + – s– s– 2 2 V := γ V = γ V : H (∂) → H (∂), (3.64) 1 1 1 + – s– s– 2 2 W := γ W + γ W : H (∂) → H (∂), (3.65) 1 3 3 + – s– s– 2 2 W := T V + T V : H (∂) → H (∂), (3.66) 1 3 + – s– s– 2 2 L := T W = T W : H (∂) → H (∂). (3.67) Mikhailov Boundary Value Problems (2018) 2018:87 Page 16 of 52 Employing relations (3.56), Theorem 3.3,and Theorem 2.6, we obtain the following mapping properties for the parametrix-based potentials on Lipschitz domains. Theorem 3.5 Let  be a bounded Lipschitz domain. 1 3 (i) The following operators are continuous if ≤ s ≤ , 2 2 s– s n s n n μV : H (∂) → H R , a ∈ C R , ∀ μ ∈ D R ; (3.68) s– s s r W : H (∂) → H (), a ∈ C (); (3.69) s– s s n μr W : H (∂) → H ( ), a ∈ C ( ), ∀ μ ∈ D R . (3.70) – – – + 1 3 2 (ii) The following operators are continuous if < s ≤ and a ∈ C (): 2 2 3 1 s– s,– 2 2 r V : H (∂) → H (; A); (3.71) s,– s– 2 n μr V : H (∂) → H ( ; A), ∀ μ ∈ D R ; (3.72) loc 1 1 s– s,– 2 2 r W : H (∂) → H (; A); (3.73) 1 1 s– s,– n 2 2 μr W : H (∂) → H ( ; A), ∀ μ ∈ D R . (3.74) 1 3 (iii) The following operators are continuous if < s < : 2 2 3 1 ± s– s– s 2 2 γ V : H (∂) → H (∂), a ∈ C ( ); (3.75) 1 1 ± s– s– s 2 2 γ W : H (∂) → H (∂), a ∈ C ( ); (3.76) 3 3 ± s– s– 2 2 T V : H (∂) → H (∂), a ∈ C ( ); (3.77) 1 3 ± s– s– 2 2 2 T W : H (∂) → H (∂), a ∈ C ( ). (3.78) + ± Proof Relations (3.56), Theorem 3.3(i), and Theorem 2.6 immediately imply the continuity of operators (3.68)and (3.69). Further, if a ∈ C (), then there exists  >0 such that a ∈ 1 3 1, + 1 3 s– 1 1 2 2 C (). For < s ≤ , g ∈ H (), and any σ ∈ ( , min{s, + }), we have 2 2 2 2 AVg σ –1 = ∇· a∇ V g = ∇· (∇ ln a)V g σ –1 H () H () a σ –1 H () ≤ (∇ ln a)V g ≤ C a V g σ H () 1, + C 2 () ≤ C a 1 V g , H () 1, + C () wherewehavetaken into accountthat V g =0 in .Hence,along with continuity σ –1 of operator in (3.58), this implies AVg ∈ H ()and thus, by Remark 2.2(4), r AVg ∈ σ –1 2 H () ⊂ H () with the corresponding norm estimate, from which the continuity of operator (3.71) follows. The continuity of operator (3.73)isprovedinasimilarfashion. The continuity of operators (3.70), (3.72), and (3.74) immediately follows from the con- tinuity of their counterparts for the interior domain. Mikhailov Boundary Value Problems (2018) 2018:87 Page 17 of 52 The continuity of operators (3.75)and (3.76) for the potential traces is implied by the continuity of operators (3.68)–(3.70) and the trace theorem, whereas the continuity of op- erators (3.77)and(3.78) for the potential co-normal derivatives is implied by the continuity of operators (3.71)–(3.74)and Theorem 2.9. Now we can prove the jump properties for the parametrix-based potentials on Lipschitz domains. 1 3 s– Theorem 3.6 Let ∂ be a compact Lipschitz boundary, < s < , ϕ ∈ H (∂), and ψ ∈ 2 2 s– H (∂). Then + – + – s n γ V ψ – γ V ψ =0, γ W ϕ – γ W ϕ =–ϕ, if a ∈ C R ; (3.79) + – + – 2 n T V ψ – T V ψ = ψ, T W ϕ – T W ϕ =(∂ a)ϕ, if a ∈ C R . (3.80) Proof Relations (3.56)and (3.62)along with Theorem 2.6 immediately imply jump rela- tions (3.79). To prove the first jump relation in (3.80), we generalise to the parametrix-based poten- s– tials the arguments from the proof of Lemma 4.1 in [11]. Let ψ ∈ H (∂). From (3.53) we obtain, in the sense of distributions, 1 ∇a ∗ ∗ ∗ AV ψ =–A P γ ψ =–γ ψ + ∇· P γ ψ a a ∗ n =–γ ψ – ∇· (∇a)V ψ in R , (3.81) ∗ ∗ wherewehavetaken into accountthat P γ ψ = γ ψ. Then, since the operator A is formally self-adjoint, for any test function φ ∈ D(R ), we obtain V ψ(y)Aφ(y) dy = AV ψ, φ n =–ψ, γφ – ∇· (∇a)V ψ , φ . (3.82) R ∂ n s– 1 3 2 n Note that, for a ∈ C (R )and ψ ∈ H (∂)with < s < , the continuity of operator 2 2 s n 2 n (3.68)and Theorem 2.6 imply that V ψ ∈ H (R )and (∇a)V ψ ∈ H (R )for some  ∈ loc loc (0, 1). Hence, from the second Green identity (2.21)with v = V ψ and u = φ,along with (3.81), we have V ψ(y)Aφ(y) dy – A V ψ, φ ± ± = V ψ(y)Aφ(y) dy – E r AV ψ, φ ± ± ± = V ψ(y)Aφ(y) dy + E r ∇· (∇a)V ψ , φ ± ± ± ± ± ± = ± T φ, γ V ψ ∓ T V ψ, γ φ . (3.83) ∂ ∂ ∗ ∗ Here we employed that r γ ψ =0 since supp γ ψ ⊂ ∂. Let us take into account + – + – c + that γ φ = γ φ = γφ and T φ = T φ = T φ due to smoothness of φ,whereas γ V ψ = Mikhailov Boundary Value Problems (2018) 2018:87 Page 18 of 52 γ V ψ = γ V ψ by the first relation in (3.79). Moreover, we also have ˚ ˚ E r ∇· (∇a)V ψ , φ = r ∇· (∇a)V ψ , E φ ± ± ± ± ± ± = ± (∂ a)γ V ψ, γφ – (∇a)V ψ, ∇φ . Then summing up (3.83)for  and  ,weobtain + – + – V ψ(y)Aφ(y) dy =– T V ψ – T V ψ, γφ + (∇a)V ψ, ∇φ . (3.84) ∂ R + – V ψ – T V ψ, γφ = ψ, γφ for arbitrary Comparing (3.84)and (3.82), we obtain T ∂ ∂ φ ∈ D(R ), which implies the first jump relation in (3.80). s– Let us similarly prove the second jump relation in (3.80). Let ϕ ∈ H (∂). From (3.54) we obtain, in the sense of distributions, 1 ∇a c∗ c∗ c∗ AW ϕ =–A P T ϕ =–T ϕ + ∇· P T ϕ a a c∗ n =–T ϕ – ∇· (∇a)W ϕ in R , (3.85) c∗ c∗ wherewehavetaken into accountthat P T ϕ = T ϕ. Then for any test function φ ∈ D(R ), we obtain c∗ W ϕ(y)Aφ(y) dy = AW ϕ, φ n =– T ϕ + ∇· (∇a)W ϕ , φ =– ϕ, T φ + (∇a)W ϕ, ∇φ . (3.86) ∂ R 1 3 2 n s– Note that for a ∈ C (R )and ϕ ∈ H (∂)with < s < , the continuity of operators 2 2 s s (3.69)and (3.70)and Theorem 2.6 imply that r W ϕ ∈ H ( ), r W ϕ ∈ H ( ), and +  – + – loc 1 + + 2 (∇a)r W ϕ ∈ H ( ), (∇a)r W ϕ ∈ H ( )forsome  ∈ (0, 1). Hence from the sec- +  – + – loc ond Green identity (2.21)for v = W ϕ and u = φ,along with (3.85), we have W ϕ(y)Aφ(y) dy – A W ϕ, φ ± ± = W ϕ(y)Aφ(y) dy – E r AW ϕ, φ ± ± ± = W ϕ(y)Aφ(y) dy + E r ∇· (∇a)W ϕ , φ ± ± ± ± ± ± = ± T φ, γ W ϕ ∓ T W ϕ, γ φ . (3.87) ∂ ∂ c∗ c∗ Here we employed that r T ϕ =0 since supp T ϕ ⊂ ∂. Let us also take into account + – + – c + that γ φ = γ φ = γφ and T φ = T φ = T φ due to smoothness of φ,whereas γ W ϕ – γ W ϕ =–ϕ by the second relation in (3.79). Moreover, we also have ˚ ˚ E r ∇· (∇a)W ϕ , φ = r ∇· (∇a)W ϕ , E φ ± ± ± ± ± ± = ± (∂ a)γ W ϕ, γφ – (∇a)W ϕ, ∇φ . ± Mikhailov Boundary Value Problems (2018) 2018:87 Page 19 of 52 Then summing up (3.87)for  and  ,weobtain + – W ϕ(y)Aφ(y) dy – (∂ a)ϕ, γφ – (∇a)W ϕ, ∇φ ∂ R c + – =– T φ, ϕ – T W ϕ – T W ϕ, γφ . (3.88) ∂ ∂ + – Comparing (3.88)and (3.86), we obtain T W ϕ – T W ϕ, γφ = (∂ a)ϕ, γφ for ar- ∂ ν ∂ bitrary φ ∈ D(R ), which implies the second jump relation in (3.80). Theorem 3.5(iii) and the first relation in (3.79) imply the following assertion. 1 3 Corollary 3.7 Let ∂ be a compact Lipschitz boundary, and let < s < . The following 2 2 operators are continuous: 3 1 + – s– s– s 2 2 V := γ V = γ V : H (∂) → H (∂), a ∈ C ( ); (3.89) 1 1 1 + – s– s– s 2 2 W := γ W + γ W : H (∂) → H (∂), a ∈ C ( ); (3.90) 1 3 3 + – s– s– 2 2 2 W := T V + T V : H (∂) → H (∂), a ∈ C ( ); (3.91) + ± 1 1 3 + – s– s– 2 2 2 L := T W + T W : H (∂) → H (∂), a ∈ C ( ). (3.92) For the case of smooth boundary, the boundary operators defined in Corollary 3.7 (see [27, Eq. (7.3)] for the fundamental solution-based potentials on Lipschitz domains) cor- respond to the boundary integral (pseudo-differential) operators of direct surface values W, and the co-normal derivatives of the single-layer potential, the double-layer potential of the single-layer potential W and of the double-layer potential (see [3, Eq. (3.6)-(3.8)]) for the parametrix-based potentials on smooth domains. See also [27, Theorems 7.3, 7.4] about integral representations on Lipschitz domains of the boundary operators associated with the layer potentials based on fundamental solutions. If a = 1, then we equip the operators defined in Corollary 3.7 with subscript .Then, under the hypotheses of Corollary 3.7,wehave(see[3, Eq. (3.10)–(3.13)] for the potentials on smooth domains) 1 1 Vg = V g, Wg = W (ag), (3.93) a a ∂ a ∂ a ν ν W g = W g – V g, Lg = L (ag)– W (ag). (3.94) a a Indeed, relations (3.93) immediately follow from (3.89), (3.90), and (3.56). Further, T Vg = + 1 T ( V g). Let {v }⊂ D()be a sequence such that v – V g 1 → 0as k →∞, k k s,– H (; which implies that also v – Vg → 0as k →∞.Then(see[32, Lemma 6.10]) s,– H (;A) 1 1 ∂ a + c c + T Vg = lim T v = lim aT v = lim ∂ v – γ v k k ν k k k→∞ a k→∞ a k→∞ a ∂ a + + = T V g – γ V g. a Mikhailov Boundary Value Problems (2018) 2018:87 Page 20 of 52 ∂ a – – ν – Similarly, T Vg = T V g – γ V g,which,togetherwith(3.91), implies the first relation in (3.94). The second relation in (3.94) is proved by similar arguments. Employing definitions (3.89)–(3.92), the jump properties (3.79)–(3.80)can be re-written 3 1 s– s– 1 3 2 2 for ψ ∈ H (∂)and ϕ ∈ H (∂)with < s < as follows: 2 2 ± ± s n γ V ψ = V ψ, γ W ϕ = ∓ ϕ + W ϕ if a ∈ C R ; (3.95) 1 1 ±  ± n T V ψ = ± ψ + W ψ, T W ϕ = ± (∂ a)ϕ + Lϕ if a ∈ C R . (3.96) ν + 2 2 4 The third Green identity and integral relations In this section, we apply some limiting procedures to obtain the parametrix-based third Green identity. 1 3 s s Theorem 4.1 Let  be a bounded Lipschitz domain, u ∈ H (), < s < , and a ∈ C (). 2 2 (i) The following generalised third Green identity holds: u + Ru + W γ u = PA u in , (4.1) where, by (2.9) and (2.10), ˇ ˇ ˇ ˚ PA u(y):= A u, P(·, y) =–E u, P(·, y) =– E (a∇u), ∇P(·, y) = ∇· P E (a∇u)(y), a.e. y ∈ , (4.2) a(y) and, particularly, if s =1, then PA u(y)=– a(x)∇u(x) ·∇ P(x, y) dx, a.e. y ∈ . (4.3) s–2 ˜ ˜ (ii) Moreover, if Au = r f in , where f ∈ H (), then the generalised third Green identity takes the form + + ˜ ˜ u + Ru – VT (f ; u)+ W γ u = Pf in . (4.4) Proof (i) Let first u ∈ D(). For y ∈ ,let B (y) ⊂  be a ball centred in y with sufficiently small radius ,and let  :=  \ B (y). For any fixed y, evidently, P(·, y)= P (·, y) ∈ a(y) 1,0 D( ) ⊂ H (A;  ) and has the coinciding classical and canonical co-normal derivatives on ∂ . Then from the first Green identity (2.20) applied to  with v = P(·, y)we obtain + + + + – T P(·, y), γ u – T P(·, y), γ u + R(·, y), u ∂B (y) ∂ =– ∇P(·, y), a∇u . (4.5) Since + + lim T P(·, y), γ u ∂B (y) →0 = lim ∂ P (x, y) a(x)γ u(x) dS(x)=–u(y), ν(x) →0 a(y) ∂B (y) Mikhailov Boundary Value Problems (2018) 2018:87 Page 21 of 52 by passing to the limits as  → 0equation (4.5) reduces to the third Green identity (4.1) for any u ∈ D(). Taking into account the denseness of D()in H () and the mapping properties of the volume potentials (3.30)and (3.35)inTheorem 3.2 and of the double- layer potential (3.69)inTheorem 3.5(i), we obtain that (4.1)–(4.2) also hold for any u ∈ s 1 3 s H ()with < s < in the sense of H (), which also implies (4.3)for s =1. 2 2 (ii) Let {u }∈ D() be a sequence converging to u in H (). By (4.2), (4.3), and (2.18) we have ˇ ˇ PA u (y)=– lim a(x)∇u (x) ·∇ P(x, y) dx =– lim E u , P(·, y) k k x  k →0 →0 = lim (A u )(x)P(x, y) dx – P(x, y)T u (x) dS(x) k k →0 ∂B (y) + + – P(x, y)T u (x) dS(x) = PA u (y)+ VT u (y). (4.6) k  k k s–2 s–2 s–2 s–2 ˜ ˜ ˇ ˜ Let now f := E r (A u – A u)+ f ,where E : H () → H ()is a (non-unique) k   k ˇ ˜ continuous extension operator, which exists by [31, Theorem 2.16]. Since r A u = r f , ˜ ˜ ˇ we obtain r f = r A u = r A u .Hence k   k   k s–2 ˜ ˇ ˇ r A u – r A u = r A (u – u) →0in H (), k     k s–2 ˜ ˜ and f → f in H ()as k →∞.Thenby (4.6), (3.53), and (2.19)weobtain + + –1 ˇ ˜ ˜ ˜ ˜ ˜ PA u = PA u + VT u = PA u + VT (f ; u )– V γ (f – A u ) k  k k  k k k k  k + + ˜ ˜ ˜ ˜ ˜ ˜ = PA u + VT (f ; u )+ P(f – A u )= VT (f ; u )+ Pf , k k k k  k k k k ∗ –1 ∗ ˜ ˜ ˜ ˜ wherewetookintoaccountthat γ (γ ) (f – A u )= f – A u by [31, Corollary 2.11] k  k k  k s–2 ˜ ˜ ˇ ˜ since f – A u ∈ H . Passing to the limits as k →∞,weobtain PA u(y)= Pf + k  k VT (f ; u), which by substitution into (4.1)gives (4.4). For some functions f , , , let us consider a more general “indirect” integral relation associated with (4.4): u + Ru – V  + W  = Pf in . (4.7) The following lemma extends Lemma 4.1 from [3], where the corresponding assertion was proved for f ∈ L (), s =1, a ∈ C (), and the infinitely smooth boundary. 3 1 1 3 s s s– s– 2 2 Lemma 4.2 Let < s < and a ∈ C (). Let u ∈ H (),  ∈ H (∂),  ∈ H (∂), 2 2 s–2 and f ∈ H () satisfy (4.7). Then Au = r fin , (4.8) + + V  – T (f ; u) – W  – γ u =0 in . (4.9) Proof Subtracting (4.7) from identity (4.1), we obtain ˇ ˜ V  – W  – γ u = P[A u – f]in . (4.10) Mikhailov Boundary Value Problems (2018) 2018:87 Page 22 of 52 Multiplying equality (4.10)by a, applying the Laplace operator , and taking into account ˜ ˇ ˜ (3.57)and (3.17), we get r f = r (A u)= Au in .Thismeans that f is an extension of the s–2 s–2 distribution Au ∈ H ()to H (), and u satisfies (4.8). Then (2.15)implies ˇ ˜ ˇ ˜ ˜ P[A u – f ](y)= A u – f , P(·, y) =– T (f ; u), P(·, y) = VT (f ; u)(y), y ∈ . (4.11) Substituting (4.11)into(4.10)leads to (4.9). 1 3 s s–1 ∇ For < s < , a ∈ C (), and g ∈ H (), let us introduce the operator A as 2 2 + A g := –∇· E (g∇a). (4.12) 1 3 Lemma 4.3 Let < s < . 2 2 |s–1|+1 (i) If a ∈ C (), then the following operator is continuous: ∇ s–1 s–2 A : H () → H (). (4.13) (ii) If a ∈ C (), then the following operator is continuous and compact: ∇ s s–2 A : H () → H (). (4.14) |s–1|+1 |s–1| Proof (i) If a ∈ C (), then ∇a ∈ C (), and by Theorem 2.6, ∇a is a multiplier in + + s–1 H (), which implies the continuity of operator (4.13). (ii) For 1 ≤ s < ,wehave s = |s –1| + 1, which by item (i) implies the continuity of operator (4.13) and thus the continuity and compactness of operator (4.14). For < s < 1, we need an estimate of the norm g∇a s–1 .First,byDefinition 2.5 the H () s 0,t t inclusion a ∈ C () implies that there exists t ∈ (s,1) such that a ∈ C ()= B ()= + ∞,∞ t t–1 F () (see, e.g., Proposition in [45, Sect. 2.1.2]) and hence ∇a ∈ F (). Then, by ∞,∞ ∞,∞ Theorem 1 from [45, Sect. 4.4.3], g∇a ≤ C ∇a g σ t–1 t–1 H () F () F () ∞,∞ 2,∞ ≤ C a g σ , ∀ σ ∈ (1 – t, s). (4.15) 0,t H () C () On the other hand, by (3.49), item (ii) of Proposition from [45, Sect. 2.2.1], and (4.15)we obtain A g ≤ c g∇a s–1 = c g∇a s–1 s–2 3 H () 3 F () H () 2,2 ≤ C g∇a t–1 ≤ C C a 0,t g . 1 1 H () F () C () 2,∞ ∇ σ s–2 Thus the operator A : H () → H () is continuous, which implies the continuity and, by the Rellich compact embedding theorem, also the compactness of operator (4.14)for < s <1. 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 23 of 52 In accordance with notation (2.10), let us also denote ˇ ˚ g := ∇· E r ∇g. Let us now discuss the trace and two forms of the co-normal derivative associated with equation (4.7). Lemma 4.4 (i) Under the hypotheses of Lemma 4.2, + + + γ u + γ Ru – V  –  + W  = γ Pf on ∂, (4.16) + + ∇ T (f ; u)+ T A u; aRu –  – W  + L (a) ˜ ˜ = T (f ; P f ) on ∂. (4.17) (ii) If, moreover, a ∈ C (), then + +  + + ˜ ˜ ˚ ˜ ˜ T (f ; u)+ T Ru –  – W  + T W  = T (f + ER f ; Pf ) on ∂, (4.18) where R is defined in (3.14) and (3.16). Proof (i) Equation (4.16)isimplied by (4.7)and (3.95). To prove (4.17), let us first multiply (4.7)by a to obtain –V  + W (a)= P f – au – aRu in . (4.19) Since {–V  +W (a)} = 0, for the both sides of (4.19) the canonical co-normal deriva- tive T is well defined, + + + –T V  + T W (a)= T (P f – au – aRu), (4.20) and by (2.17) + –1 ˜ ˜ T (P f – au – aRu)=– γ (P f – au – aRu), (4.21) because by (4.19), ˜ ˜ ˚ ˜ ˚ (P f – au – aRu)= E (P f – au – aRu)= E – V  + W (a) =0. Note that, by the second equality in (3.16), (aRu)=–∇· P (u∇a) =–∇· (u∇a)= r A u in , (4.22) ∇ s–2 s–2 which implies that A u ∈ H () is an extension of (aRu) ∈ H (). Further (see (2.10)), ˇ ˚ ˚ ˚ (au)= ∇· E r ∇(ua)= ∇· E r (u∇a)+ ∇· E r (a∇u) Mikhailov Boundary Value Problems (2018) 2018:87 Page 24 of 52 ∇ n =–A u + A u in R . Then ∇ n ˇ ˜ ˜ ˇ ˜ ˜ ˇ ˇ (P f – au – aRu)= f – P f –(f – A u)– A u – (aRu) in R , and by (2.13) + + + + ∇ ˜ ˜ ˜ ˜ T (P f – au – aRu)= T (f ; P f )– T (f ; u)– T A u; aRu . Substituting this in (4.20), we obtain + + ∇ + + + ˜ ˜ ˜ T (f ; u)+ T A u; aRu – T V  + T W (a)= T (f ; P f)on ∂. Taking into account jump relation (3.63)and (3.66)with(3.67), we arrive at (4.17). (ii) To prove (4.18), letusfirstremarkthat ˜ ˜ ˜ APf = f + R f in , (4.23) ˜ ˜ which implies, due to (4.8), that A(Pf – u)= R f in ,where R is defined in (3.14)and ∗ ∗ 2 σ 1 (3.16), and since a ∈ C (), we obtain by (3.39)that R f ∈ H ()for some σ >– .Then + ∗ σ s–2 ˜ ˜ ˜ ˚ ˜ A(Pf – u) can be canonically extended to A(Pf – u)= E R f ∈ H () ⊂ H (). This implies that there exists a canonical co-normal derivative of (Pf – u), for which, due to (2.17)and (2.13), we have + –1 ˜ ˜ ˜ ˇ ˜ ˇ T (Pf – u)= γ A(Pf – u)– A Pf + A u –1 ˚ ˜ ˇ ˜ ˇ = γ [E R f – A Pf + A u] –1 ˜ ˇ ˜ ˇ ˜ ˚ ˜ = γ [f + E R f – A Pf + A u – f ] + + ˜ ˜ ˜ ˜ = T (f + E R f , Pf )– T (f , u), (4.24) s–2 ˜ ˚ ˜  ˜ where f + E R f ∈ H () is an extension of APf due to (4.23). From (4.7)wehave Pf – u = Ru – V  + W  in . Substituting this in the left-hand side of (4.24) and taking into account jump relation (3.96), we arrive at (4.18). Note that, unlike (4.17), the co-normal derivative form (4.18)ofrelation(4.7)iswritten without referring to the corresponding constant-coefficient potentials. 1 3 –1/2 s–2 Remark 4.5 Let < s < and f ∈ H () ⊂ H (). 2 2 s,–1/2 (i) Then evidently P f ∈ H (, ) and + + ˜ ˜ ˜ T (f ; P f )= T P f . (4.25) –1/2 (ii) Furthermore, if the hypotheses of Lemma 4.2 aresatisfiedand f ∈ H (),then s,–1/2 + + + ˜ ˜ (4.8)implies that u ∈ H (, A) and T (f ; u)= T (Au; u)= T u.Henceforth, (4.17) takes the simpler form + + ∇  + + T u + T A u; aRu –  – W  + L (a)= T P f on ∂. (4.26) 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 25 of 52 s,–1/2 If, in addition, au ∈ H (, ),thenby (4.22) ∇ 2 (aRu)= r A u =–∇· (u∇a)= Au – (au) ∈ H (). Hence the canonical co-normal derivative T (aRu) is well defined, and by (2.13), (2.17), (3.16), and (4.22) + ∇ T A u; aRu –1 ∇ = γ A u – (aRu) –1 ∇ + = γ A u – (aRu) + T (aRu) –1 + ˚ ˚ = γ –∇· E (u∇a)+ E ∇· (u∇a) + T (aRu) ∗ ∗ –1 –1 + ˇ ˜ ˇ ˜ = γ [Au – Au]+ γ – (au)+ (au) + T (aRu) + + + =–T u + T (au)+ T (aRu). (4.27) This reduces (4.26)tothe relation + +  + + T (au)+ T (aRu)–  – W  + L (a)= T P f on ∂ (4.28) with only canonical normal derivatives associated with the Laplace operator involved. –1/2 (iii) If the hypotheses of Lemma 4.2 aresatisfiedand,moreover, f ∈ H (),and 2 s,– –1/2 ˜ ˜ a ∈ C (), then, by (3.32)and (3.39), Pf ∈ H (; A) and R f ∈ H (), + ∗ + + ˜ ˚ ˜ ˜ ˜ implying T (f + E R f ; Pf )= T (Pf ).Henceforth, (4.18) reduces to the relation + +  + + T u + T Ru –  – W  + T W  = T Pf on ∂ with only canonical co-normal derivatives associated with the operator A involved. Remark 4.6 (i) Let the hypotheses of Lemma 4.2 be satisfied and suppose that a sequence – s–2 ˜ ˜ {f }∈ H ()converges to f in H (). By the continuity of operators (3.30)and (3.34), estimate (2.14), and relation (4.25)for f ,weobtainthat + + s– ˜ ˜ ˜ T (f ; P f )= lim T P f in H (∂); j→∞ see also Theorem 7.1. (ii) If, moreover, a ∈ C (), then, similarly, + + + ˜ ˚ ˜ ˜ ˜ ˚ ˜ ˜ ˜ T (f + E R f , Pf )= lim T (f + E R f , Pf )= lim T Pf . ∗ j  ∗ j j j j→∞ j→∞ Lemma 4.4 and the third Green identity (4.4) imply the following assertion. 1 3 s Corollary 4.7 Let  be a bounded Lipschitz domain, and let < s < , a ∈ C (), u ∈ 2 2 + s s–2 ˜  ˜ H (), and f ∈ H () be such that Au = r fin . Mikhailov Boundary Value Problems (2018) 2018:87 Page 26 of 52 (i) Then + + + + + ˜ ˜ γ u + γ Ru – VT (f ; u)+ W γ u = γ Pf on ∂, (4.29) + + ∇  + + ˜ ˜ T (f , u)+ T A u; aRu – W T (f , u)+ L aγ u ˜ ˜ = T (f ; P f ) on ∂. (4.30) (ii) If, moreover, a ∈ C (), then + +  + + + ˜ ˜ T (f , u)+ T Ru – W T (f , u)+ T W γ u ˜ ˚ ˜ ˜ = T (f + E R f , Pf ) on ∂, (4.31) where R is defined in (3.14) in (3.16). 1 3 Let us extend to Lipschitz domains and s ∈ ( , ) Lemma 4.2(i,ii) from [3], which is 2 2 proved there for smooth domains and s =1. Lemma 4.8 Let  be a bounded simply connected Lipschitz domain, and let a ∈ C () 1 3 with < s < . 2 2 ∗ s– ∗ ∗ (i) If  ∈ H (∂) and r V  =0, then  =0. ∗ s– ∗ ∗ (ii) If  ∈ H (∂) and r W  =0, then  =0. Proof To prove (i), let us multiply equation r V  =0 by a,which by thefirstrelationin (3.56)reduces it to r V  =0 in . Taking the trace of this equation on ∂ and using the first relation in (3.95)(forthe case a =1), by Theorem 7.3 we obtain item (i). Similarly, multiplying the equation r W  =0 by a, the second relation in (3.56)re- duces it to r W (a )=0 in . Taking the trace of this equation on ∂ and using the 1 ∗ ∗ ˆ ˆ first jump relation in (3.95)(forthe case a =1), we obtain –  + W  =0 on ∂,where ∗ ∗ ∗ ˆ ˆ = a .Since this equation for  is uniquely solvable (see Theorem 7.3), by condition (2.5) this implies item (ii). Theorem 4.9 Let  be a bounded simply connected Lipschitz domain, and let a ∈ C () 1 3 s–2 s ˜  ˜ with < s < . Let f ∈ H (). A function u ∈ H () is a solution of PDE Au = r fin  if 2 2 and only if it is a solution of boundary-domain integro-differential equation (4.4). Proof If u ∈ H ()solves PDE Au = r f in , then by Theorem 4.1(ii) it satisfies (4.4). On the other hand, if u solves the boundary-domain integro-differential equation (4.4), then + + using Lemma 4.2 for  = T (f ; u)and  = γ u completes the proof. 5 Segregated BDIE systems for the Dirichlet problem 1 3 For < s < , let us consider the Dirichlet problem: 2 2 Find a function u ∈ H () satisfying the equations Au = f in , (5.1) γ u = ϕ on ∂, (5.2) s–2 s– where f ∈ H () and ϕ ∈ H (∂). 0 Mikhailov Boundary Value Problems (2018) 2018:87 Page 27 of 52 Equation (5.1) is understood in the distributional sense (2.7), and the Dirichlet boundary condition (5.2) is understood in the trace sense. The following uniqueness assertion is well known for s = 1 and follows from the first Green identity; hence it also holds for 1 ≤ s < 3/2. |s–1| 3 Theorem 5.1 Let a ∈ C () with 1 ≤ s < . The Dirichlet problem (5.1)–(5.2) has at most one solution in H (). 5.1 BDIE formulations and equivalence to the Dirichlet problem 1 3 Let < s < . In this section, we reduce the Dirichlet problem (5.1)–(5.2) to three differ- 2 2 ent segregated boundary-domain integral equation (BDIE) systems. Two of these formu- lations, for s = 1 and infinitely smooth coefficients and infinitely smooth boundary, were analysed in [33]. s–2 s–2 ˜  ˜ Let f ∈ H () be an extension of f ∈ H ()(i.e., f = r f ), which always exists; see [31, Lemma 2.15 and Theorem 2.16]. Let us substitute into (4.4), (4.29), (4.30), and (4.31) the generalised co-normal derivative and the trace of the function u as + + T (f ; u)= ψ, γ u = ϕ , where ϕ is the known right-hand side of the Dirichlet boundary condition (5.2), and ψ ∈ s– H (∂) is a new unknown function that will be regarded as formally segregated from u. s s– Thus we will look for the unknown couple (u, ψ) ∈ H () × H (∂). BDIE system (D1). Let a ∈ C (). To reduce the Dirichlet BVP (5.1)–(5.2)tothe BDIE system (D1), we will use equation (4.4)in  and equation (4.29)on ∂. Then we arrive at the following system of the boundary-domain integral equations, (D1), which is similar to the corresponding system in [33]: D1 u + Ru – V ψ = F in , (5.3) + D1 γ Ru – V ψ = F on ∂, (5.4) where D1 D F F 1 0 D1 D F = = and F := Pf – W ϕ in . (5.5) D1 D F γ F – ϕ 2 0 s– s–2 D s Note that, for ϕ ∈ H (∂)and f ∈ H (), we have the inclusion F ∈ H ()due to the mapping properties of the Newtonian (volume) and layer potentials; see (3.30)and D1 s s– (3.69). Hence F ∈ H () × H (∂). BDIE system (D2 ). Let a ∈ C (). To obtain a segregated BDIE system of the second kind,wewilluse equation (4.4)in  and equation (4.30)on ∂. Then we arrive at the following BDIE system (D2 ): D2 u + Ru – V ψ = F in , (5.6) + ∇  D2 ψ + T A u; aRu – W ψ = F on ∂, (5.7) 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 28 of 52 where D2 F Pf – W ϕ D2 F = = . (5.8) D2 ˜ ˜ F T (f ; P f )– L (aϕ ) D2 Due to the mapping properties of the operators involved in (5.11), we have F ∈ s s– H () × H (∂). BDIE system (D2). Let the coefficient be smoother than in the first two cases, a ∈ C (). Now we will use equation (4.4)in  and equation (4.31)on ∂. Then we arrive at another BDIE system of the second kind, (D2), which is similar to the corresponding system in [33]: D2 u + Ru – V ψ = F in , (5.9) +  D2 ψ + T Ru – W ψ = F on ∂, (5.10) where D2 F Pf – W ϕ D2 1 F = = . (5.11) D2 + + ˜ ˚ ˜ ˜ F T (f + E r R f ; Pf )– T W ϕ ∗ 0 D2 s Due to the mapping properties of the operators involved in (5.11), we have F ∈ H () × s– H (∂). LetusprovethatBVP(5.1)–(5.2)in  is equivalent to each of the three systems of BDIEs, (D1), (D2 ), and (D2). s 1 3 s– s–2 Theorem 5.2 Let a ∈ C () with < s < . Let ϕ ∈ H (∂), f ∈ H (), and f ∈ 2 2 s–2 H () be such that r f = f . (i) If a function u ∈ H () solves the Dirichlet BVP (5.1)–(5.2), then the couple s s– (u, ψ) ∈ H () × H (∂), where ψ = T (f ; u) on ∂ (5.12) solves the BDIE systems (D1), (D2 ) and, if a ∈ C (), then also the BDIE system (D2). s s– (ii) Vice versa, if a a couple (u, ψ) ∈ H () × H (∂) solves one of the BDIE systems, (D1), (D2 ), or (D2) (if a ∈ C ()), then this solution solves the other BDIE systems, whereas u solves the Dirichlet BVP, and ψ satisfies (5.12). Proof (i) Let u ∈ H ()be a solution to BVP (5.1)–(5.2). Setting ψ by (5.12) evidently im- s– plies ψ ∈ H (∂). Then it immediately follows from Theorem 4.9 and relations (4.29)– (4.31) that the couple (u, ψ)solves systems (D1), (D2) ,and,if a ∈ C (), then also (D2), with the right-hand sides (5.5), (5.8), and (5.11), respectively, which completes the proof of item (i). s s– (ii) Let now a couple (u, ψ) ∈ H () × H (∂)solve BDIE system (5.3)–(5.4). Taking the trace of equation (5.3)on ∂, and subtracting equation (5.4)fromit, we obtain γ u = ϕ on ∂, (5.13) 0 Mikhailov Boundary Value Problems (2018) 2018:87 Page 29 of 52 i.e., u satisfies the Dirichlet condition (5.2). Equation (5.3) and Lemma 4.2 with  = ψ and = ϕ imply that u is a solution of PDE (5.1), and ∗ ∗ V  – W  =0 in , ∗ + ∗ + ∗ where  = ψ – T (f ; u)and  = ϕ – γ u.Due to equation (5.13),  =0. Then Lemma 4.8(i) implies  = 0, i.e., condition (5.12). Thus u obtained from solution of BDIE system (D1) solves the Dirichlet problem and hence, by item (i) of the theorem, (u, ψ) solves also BDIE system (D2 )and,if a ∈ C (), then also (D2). s s– Let now a couple (u, ψ) ∈ H () × H (∂)solve BDIE system (5.6)–(5.7). Lemma 4.2 for equation (5.6)implies that u is a solution of PDE (5.1), and equation (4.9)holds for = ψ and  = ϕ , whereas Corollary 4.7 gives equation (4.30). Multiplication of (4.9)by a reduces it to + + V ψ – T (f ; u) – W a ϕ – γ u =0 in . (5.14) Subtracting (4.30)from equation(5.7) and taking into account (5.14)give ψ – T (f ; u)=0 on ∂, (5.15) that is, equation (5.12)isproved. Equations(5.14)and (5.15)give W  =0 in ,where ∗ + ∗ = a(ϕ – γ u). Then Lemma 4.8(ii) implies  =0 on ∂.Thismeans that u satisfies the Dirichlet condition (5.2). Thus u obtained from solution of BDIE system (D2 )solves the Dirichlet problem, and hence, by item (i) of the theorem, the couple (u, ψ)solves also BDIE system (D1) and, if a ∈ C (), then also (D2). 2 s s– Let, finally, a ∈ C (), and let a couple (u, ψ) ∈ H () × H (∂)solve BDIE system (5.9)–(5.10). Lemma 4.2 for equation (5.9)implies that u is a solution of PDE (5.1), and equation (4.9)holds for  = ψ and  = ϕ , whereas Corollary 4.7 gives equation (4.31). Subtracting (4.31)fromequation(5.10) and adding to it the canonical co-normal deriva- + ∗ tive T of equation (4.9)leadto(5.12). Equations (4.9)and (5.12)imply W  =0 in , ∗ + ∗ where  = ϕ – γ u. Then by Lemma 4.8(ii) we deduce  =0 on ∂.Thismeans that u satisfies the Dirichlet condition (5.2). Thus u obtained from solution of BDIE system (D2) solves the Dirichlet problem, and hence, by item (i) of the theorem, the couple (u, ψ) solves also BDIE systems (D1) and (D2 ). 5.2 Properties of BDIE system operators for the Dirichlet problem BDIE systems (D1), (D2 ), and (D2) can be written as 1 D D1 2 D D2 2 D D2 D U = F , D U = F ,and D U = F , D  s s– respectively. Here U := (u, ψ) ∈ H () × H (∂), I – R –V D := , γ R –V I + R –V I + R –V D := , D := , 1 1 + ∇  + T (A ; aR) I – W T R I – W 2 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 30 of 52 D1 D2 D2 whereas F , F ,and F are given by (5.5), (5.8), and (5.11), respectively. Note that + ∇ –1 ∇ T A ; aR u := γ A u – (aRu) . (5.16) 1 3 Let < s < . The operators 2 2 3 1 1 s s– s s– s 2 2 D : H () × H (∂) → H () × H (∂)if a ∈ C (), (5.17) 1 3 s s– s s– s 2 2 D : H () × H (∂) → H () × H (∂)if a ∈ C (), (5.18) 1 3 2 s s– s s– 2 2 2 D : H () × H (∂) → H () × H (∂)if a ∈ C (), (5.19) are continuous due to the mapping properties of the operators constituting them (see Sect. 3), whereas for the right-hand sides of the BDIE systems, we have the inclusions 1 3 3 D1 s s– D2 s s– D2 s s– 2 2 2 F ∈ H () × H (∂), F ∈ H () × H (∂), and F ∈ H () × H (∂). 1 3 Theorem 5.3 Let  be a bounded simply connected Lipschitz domain, and let < s < . 2 2 Operators (5.17)–(5.19) are Fredholm operators with zero index. Proof The continuity of operators has been already proved. To prove the Fredholm property of operator (5.17), let us consider the operator 3 1 I –V 1 s s– s s– 2 2 D := : H () × H (∂) → H () × H (∂). (5.20) 0–V As a result of compactness properties of the operators R and γ R given by (3.35)and (3.42)inTheorem 3.2,operator(5.20) is a compact perturbation of operator (5.17). The operator D is an upper triangular matrix operator with the following scalar diagonal in- vertible operators: s s I : H () → H (), 3 1 s– s– 2 2 V : H (∂) → H (∂), where the invertibility of the operator V is implied by the invertibility of operator V in (7.4)and by thefirstrelationin(3.93). This implies that operator (5.20) is invertible. Thus (5.17) is a Fredholm operator with zero index. The operator I –V 3 3 2 s s– s s– 2 2 D := : H () × H (∂) → H () × H (∂) (5.21) 0 I – W s s is a compact perturbation of operator (5.18). Indeed, the operators R : H () → H () + ∇ s is compact due to Theorem 3.2. The compactness of the operator T (A ; aR): H () → s– ∇ s s–2 H (∂), defined by (5.16), follows from that of the operator A : H () → H ()given s s by Lemma 4.3(ii) and of the operator R : H () → H (), i.e., operator (3.35)inTheo- rem 3.2. Mikhailov Boundary Value Problems (2018) 2018:87 Page 31 of 52 Consider the diagonal operators of the upper triangular matrix operator D .The op- s s erator I : H () → H () is evidently invertible, whereas the invertibility of the operator 3 3 s– s– 2 2 I – W : H (∂) → H (∂)is stated by Theorem 7.3. This implies that operator (5.21) is invertible, and hence operator (5.18) is Fredholm with zero index. Operator (5.21)isalsoacompactperturbationofoperator(5.19). Indeed, the opera- s s + s s– tors R : H () → H ()and T R : H () → H (∂) are compact due to Theorem 3.2. 2  ∂ a From the first representation in (3.94), for a ∈ C (), the operator W – W = V : s– σ 1 1 H (∂) → H (∂), where σ = min{ , s – }, is continuous, which implies that the op- 2 2 3 3 s– s– 2 2 erator W – W : H (∂) → H (∂) is compact. Since operator (5.21) is invertible, this implies that operator (5.19) is Fredholm with zero index. 1 3 Theorem 5.4 Let  be a bounded simply connected Lipschitz domain, < s < , and σ = 2 2 max{1, s}. The following operators are continuously invertible: 3 1 1 s s– s s– σ 2 2 D : H () × H (∂) → H () × H (∂) if a ∈ C (), (5.22) 1 3 s s– s s– σ 2 2 D : H () × H (∂) → H () × H (∂) if a ∈ C (), (5.23) 1 3 2 s s– s s– 2 2 D : H () × H (∂) → H () × H (∂) if a ∈ C (). (5.24) Proof First, let 1 ≤ s < .Then σ = s, and the injectivity of operators (5.22)–(5.24)isim- plied by the equivalence Theorem 5.2(ii) and the BVP uniqueness Theorem 5.1. Indeed, consider, for example, the injectivity of operator (5.22). For the homogeneous equation 1 D D1 D U = 0, its zero right-hand side F =0 can be represented as in (5.5)interms of f =0 D + and ϕ =0. Then, by Theorem 5.2(ii), U =(u, T (0; u)) ,where u is a solution of the Dirichlet problem (5.1)–(5.2) with the right-hand sides f =0 and ϕ = 0, which has only the trivial solution u =0 due to Theorem 5.1. The arguments for the injectivity of opera- tors (5.23)and (5.22)are similar. Since, by Theorem 5.3, operators (5.22)–(5.24) are Fredholm with zero index, this im- plies their invertibility for 1 ≤ s < . 1 2 Let now < s ≤ 1. Then σ = 1, i.e., a ∈ C () for operators (5.22)–(5.23), and a ∈ C () for operator (5.24). Hence, for a fixed function a satisfying the corresponding conditions in (5.22)–(5.23), all these operators are continuous for < s ≤ 1. By Theorem 5.3 they are also Fredholm with zero index. Since, as already proved, at s = 1, these operators are also invertible, Lemma 7.5 implies that their kernels (null-spaces) consist of only the zero element for any s ∈ ( , 1], which implies that the operators are invertible for all s from this interval. Theorems 5.4 and 5.2 imply the following assertion. 1 3 Corollary 5.5 Let  be a bounded simply connected Lipschitz domain, < s < , f ∈ 2 2 s–2 s– σ H (), ϕ ∈ H (∂), and a ∈ C () with σ = max{1, s}. Then the Dirichlet problem s D –1 (5.1)–(5.2) is uniquely solvable in H (). The solution is u =(A ) (f , ϕ ) , where the s– D –1 s–2 s inverse operator (A ) : H () × H (∂) → H () to the left-hand side operator s– D s s–2 A : H () → H () × H (∂) of the Dirichlet problem (5.1)–(5.2) is continuous. s–2 s–2 Remark 5.6 For a given function f ∈ H (), its extension f ∈ H () is not unique. Nevertheless, since the solution of the Dirichlet BVP (5.1)–(5.2) does not depend on this Mikhailov Boundary Value Problems (2018) 2018:87 Page 32 of 52 extension, equivalence Theorem 5.2(ii) implies that u in the solution of BDIE systems (D1) and (D2) does not depend on the particular choice of extension f although ψ obviously does; see (5.12). 6 Segregated BDIE systems for the Neumann problem Let us consider the Neumann problem: Find a function u ∈ H () satisfying the equations Au = r f in , (6.1) T (f ; u)= ψ on ∂, (6.2) s– s–2 where ψ ∈ H (∂)and f ∈ H (). Equation (6.1) in understood in the distribution sense (2.7), and the Neumann bound- ary condition (6.2)inthe sense(2.13). The following assertion is well known and can be proved, e.g., using variational settings and the Lax–Milgram lemma. Theorem 6.1 Let s =1 and a ∈ L (). (i) The homogeneous Neumann problem (6.1)–(6.2) admits only one linearly 0 1 independent solution u =1 in H (). (ii) The non-homogeneous Neumann problem (6.1)–(6.2) is solvable if and only if 0 + 0 f , u – ψ , γ u = 0. (6.3) |s–1| Remark 6.2 Item (i) in Theorem 6.1 evidently implies that, for 1 ≤ s < and a ∈ C (), the homogeneous Neumann problem associated with (6.1)–(6.2) also admits only one lin- 0 s early independent solution u =1 in H (). 6.1 BDIE formulations and equivalence to the Neumann problem 1 3 Let < s < . We will explore different possibilities of reducing the Neumann problem 2 2 (6.1)–(6.2) to a BDIE system. Let us represent in (4.4), (4.29), (4.30), and (4.31)the gener- alised co-normal derivative and the trace of the function u as + + T (f ; u)= ψ , γ u = ϕ, where ψ is the known right-hand side of the Neumann boundary condition (6.2), and s– ϕ ∈ H (∂) is a new unknown function that will be regarded as formally segregated s s– from u. Thus we will look for the unknown couple (u, ϕ) ∈ H () × H (∂). BDIE system (N1 ). Let a ∈ C (). Using equation (4.4)in  and equation (4.30)on ∂, we arrive at the following BDIE system (N1 ) of two equations for the couple of unknowns (u, ϕ): N1 u + Ru + W ϕ = F in , (6.4) + ∇ N1 T A u; aRu + L (aϕ)= F on ∂, (6.5) where N1 F Pf + V ψ N1 F = = . (6.6) N1 ˜ ˜ F T (f ; P f )– ψ + W ψ 0 0 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 33 of 52 N1 Due to the mapping properties of the operators involved in (6.9), we have F ∈ H () × s– H (∂). BDIE system (N1). Let the coefficient be smoother than in the previous case, a ∈ C (). Now, using equation (4.4)in  and equation (4.31)on ∂, we arrive at the following BDIE system (N1) of two equations for the couple of unknowns (u, ϕ), which is similar to the corresponding system in [33]: N1 u + Ru + W ϕ = F in , (6.7) + + N1 T Ru + T W ϕ = F on ∂, (6.8) where N1 F Pf + V ψ N1 1 F = = . (6.9) N1 + 1 ˜ ˜ ˜ F T (f + E r R f ; Pf )– ψ + W ψ ∗ 0 0 N1 s Due to the mapping properties of the operators involved in (6.9), we have F ∈ H () × s– H (∂). BDIE system (N2). Let again a ∈ C (). If we use equation (4.4)in  and equation (4.29) on ∂, we arrive for the couple (u, ϕ) at the following BDIE system (N2) of two equations of the second kind, which is also similar to the corresponding system in [33]: N2 u + Ru + W ϕ = F in , (6.10) + N2 ϕ + γ Ru + W ϕ = F ,on ∂. (6.11) where N2 N F F N2 1 0 N F = = , F := Pf + V ψ in . (6.12) N2 + N F γ F 2 0 N2 s Due to the mapping properties of the operators involved in (6.12), we have F ∈ H () × s– H (∂). 1 3 s s– s–2 Theorem 6.3 Let < s < , a ∈ C (), ψ ∈ H (∂), and f ∈ H (). 2 2 (i) If a function u ∈ H () solves the Neumann problem (6.1)–(6.2), then the couple + s– 2 (u, ϕ) with ϕ = γ u ∈ H (∂) solves BDIE systems (N1 ), (N2), and, if a ∈ C (), also (N1). s s– (ii) Vice versa, if a couple (u, ϕ) ∈ H () × H (∂) solves one of the BDIE systems, (N1 ), (N2), or (N1) (if a ∈ C ()), then the couple solves the other two BDE systems, whereas u solves the Neumann problem (6.1)–(6.2) and γ u = ϕ. Proof (i) Let u ∈ H () be a solution of the Neumann problem (6.1)–(6.2). Then from Theorem 4.9 and relations (4.29)–(4.31) we see that the couple (u, ϕ)with ϕ = γ u solves BDIE systems (N1 ), (N2), and (N1) with the right-hand sides (6.6), (6.12), and (6.9), re- spectively, which proves item (i). s s– (ii) Let a couple (u, ϕ) ∈ H () × H (∂)solve BDIE system (N1 ). Lemma 4.2 for equation (6.4)implies that u is a solution of PDE (6.1), and equation (4.9)holds for  = ψ 0 Mikhailov Boundary Value Problems (2018) 2018:87 Page 34 of 52 and  = ϕ, whereas Corollary 4.7 gives equation (4.31). Multiplication of (4.9)by a reduces it to + + V ψ – T (f ; u) – W a ϕ – γ u =0 in . (6.13) Subtracting (4.31)fromequation(6.5), we get T (f ; u)= ψ on ∂, i.e., u satisfies the Neu- mann condition (6.2). Further, from (6.13)wederive W (a(ϕ – γ u)) = 0 in ,whence γ u = ϕ on ∂ by Lemma 4.8, completing item (ii) for BDIE system (N1 ). Let a couple (u, ϕ) ∈ H () × H (∂) solve BDIE system (N1). Lemma 4.2 for equation (6.7)implies that u is a solution of PDE (6.1), and equation (4.9)holds for  = ψ and = ϕ, whereas Corollary 4.7 gives equation (4.31). Subtracting (4.31)from equation(6.8) gives T (f ; u)= ψ on ∂, i.e., u satisfies the Neumann condition (6.2). Further, from (4.9) + + we derive W(γ u – ϕ)=0 in ,whence γ u = ϕ on ∂ by Lemma 4.8,completingitem (ii) for BDIE system (N1). Let now a couple (u, ϕ) ∈ H () × H (∂) solve BDIE system (N2). Further, taking the trace of (6.10)on ∂ and comparing the result with (6.11), we easily derive that γ u = ϕ on ∂. Lemma 4.2 for equation (6.10)implies that u is a solution of PDE (6.1), and equations (4.9)holds for  = ψ and  = ϕ.Further,from(4.9) and relation γ u = ϕ we derive V ψ – T (f ; u) =0 in , whence T (f ; u)= ψ on ∂ by Lemma 4.8, i.e., u solves the Neumann problem (6.1)–(6.2), which completes the proof of item (ii) for BDIE system (N2). 6.2 Properties of BDIE system operators for the Neumann problem BDIE systems (N1 ), (N1), and (N2) can be written, respectively, as N N1 1 N N1 2 N N2 N U = F , N U = F , N U = F , N  s s– where U =(u, ϕ) ∈ H () × H (∂), I + R W I + R W N := , N := , + ∇ + + T (A ; aR) L T R T W I + R W N := , + 1 γ R I + W 1 3 and we denoted L g := L (ag). Let < s < . Due to the mapping properties of the poten- 2 2 tials (see Sect. 3), the operators 1 3 s s– s s– s 2 2 N : H () × H (∂) → H () × H (∂)if a ∈ C (), (6.14) 1 3 1 s s– s s– 2 2 2 N : H () × H (∂) → H () × H (∂)if a ∈ C (), (6.15) 1 1 2 s s– s s– s 2 2 N : H () × H (∂) → H () × H (∂)if a ∈ C () (6.16) are continuous, whereas for the right-hand sides of the BDIE systems, we have the inclu- 3 3 1 N1 s s– N1 s s– N2 s s– 2 2 2 sions F ∈ H () × H (∂), F ∈ H () × H (∂), F ∈ H () × H (∂). Mikhailov Boundary Value Problems (2018) 2018:87 Page 35 of 52 1 3 Theorem 6.4 Let  be a bounded simply connected Lipschitz domain, and let < s < . 2 2 Operators (6.14)–(6.16) are Fredholm operators with zero index. Proof The continuity of operators is already proved. Let us consider operator (6.14). Due to estimate (2.5)and Theorem 7.3, the operator L : 1 3 s– s– 2 2 H (∂) → H (∂) is a Fredholm operator with zero index. Therefore the operator IW 1 3 1 s s– s s– 2 2 N := : H () × H (∂) → H () × H (∂) (6.17) 0 L is also Fredholm with zero index. Operator (6.14)isacompactperturbationof N since the operators s s R : H () → H (), (6.18) + ∇ s s– T A ; aR : H () → H (∂) (6.19) are compact due to Theorem 3.2,ashas been showninthe compactness proofrelated to operator (5.21). Thus operator (6.14) is Fredholm with zero index as well. Operator (6.17) is also a compact perturbation of operator (6.15). Indeed, the operators (6.18), 1 3 + s– s– 2 2 T W – L : H (∂) → H (∂), + s s– T R : H () → H (∂) are compact, due to relations (3.94)and (3.96)and Theorem 3.7.Thusoperator(6.15)is Fredholm with zero index as well. To analyse operator (6.16), let us consider the auxiliary operator 1 1 IW 2 s s– s s– 2 2 N := : H () × H (∂) → H () × H (∂). (6.20) 0 I + W 1 1 1 For any function g,wecan represent ( I + W)g = ( I + W )(ag), which, by Theorem 7.3, 2 a 2 1 1 1 s– s– 2 2 implies that the operator I + W : H (∂) → H (∂) and hence operator (6.20)are Fredholm with zero index. Due to the compactness of operator (6.18), operator (6.16)isa compact perturbation of operator (6.20) and thus is Fredholm with zero index as well. 1 3 Theorem 6.5 Let  be a bounded simply connected Lipschitz domain, < s < , and σ = 2 2 max{1, s}. The following operators have one-dimensional null-spaces, ker N = ker N = s– 2 s 0 0 ker N , in H () × H (∂), spanned over the element (u , ϕ )=(1,1): 1 3 s s– s s– σ 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (), (6.21) 1 3 1 s s– s s– 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (), (6.22) 1 1 2 s s– s s– σ 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (). (6.23) + Mikhailov Boundary Value Problems (2018) 2018:87 Page 36 of 52 Proof The conditions on the coefficient a imply that, for s = 1, operators (6.21)–(6.23)are continuous. Then the equivalence Theorem 6.3 and Theorem 6.1(i) imply that the homo- geneous BDIE systems (N1 ), (N1), and (N2) have only one linear independent solution 0 0 0   1 U =(u , ϕ ) = (1, 1) in H () × H (∂). Indeed, consider, for example, the homoge- N N1 neous equation N U = 0. Its zero right-hand side F =0 can be represented as in N + (6.6)interms of f =0 and ψ =0. Then, by Theorem 6.3(ii), U =(u, γ u) ,where u is a solution of the Neumann problem (6.1)–(6.2) with the right-hand sides f =0 and ψ =0, which has only the one linearly independent solution, u =1, due to Theorem 6.1.This proves the theorem for s = 1, and then Lemma 7.5 and Theorem 6.4 complete the proof 1 3 for < s < . 2 2 1 3 Lemma 6.6 Let  be a bounded simply connected Lipschitz domain, < s < , and 2 2 σ s s– a ∈ C () with σ = max{1, s}. For any couple (F , F ) ∈ H () × H (∂), there exists 1 2 s– s–2 auniquecouple (f ,  ) ∈ H () × H (∂) such that ∗ ∗ F = Pf – W  in , (6.24) 1 ∗ ∗ ˜ ˜ F = T (f ; Pf )– L (a ) on ∂. (6.25) 2 ∗ ∗ 3 1 s s– s–2 s– 2 2 Moreover,(f ,  )= C (F , F ), and C : H () × H (∂) → H () × H (∂) is a ∗ ∗ ∗ 1 2 ∗ linear continuous operator given by ˜ ˇ f = (aF )+ γ F , (6.26) ∗  1 2 –1 1 1 + ∗ = – I + W γ –aF + P (aF )+ γ F . (6.27) 1 2 a 2 s–2 s– Proof Let us first assume that there exist (f ,  ) ∈ H () × H (∂)satisfying equa- ∗ ∗ tions (6.24)and (6.25) and find their expressions in terms of F and F . Multiplying (6.24) 1 2 by a,weget aF – P f =–W (a )in . (6.28) Applying the Laplace operator to (6.28), we obtain ˜ ˜ (aF – P f )= (aF )– f =– W (a )=0 in , (6.29) ∗ 1 ∗ which means (aF )= r f in  (6.30) 1  ∗ s,0 + and aF – P f ∈ H (; ). Applying the canonical co-normal derivative operator T to ˜ ˜ ˜ both sides of equation (6.28) and taking into account that – W (a )= (aF – P f )= ∗ 1 0because W (a ) is a harmonic function in ,weobtain, dueto(2.17)and (2.13), –L (a )=–T W (a ) = T (aF – P f ) Mikhailov Boundary Value Problems (2018) 2018:87 Page 37 of 52 –1 ˜ ˜ ˜ ˇ = γ (aF – P f )– (aF – P f ) ∗  1 –1 + ˜ ˜ =– γ (aF – P f )= T (0; aF – P f ), (6.31) ∗ 1 where (6.30) was taken into account. Substituting this into (6.25), we obtain F = T (f , aF )on ∂. (6.32) 2 ∗ 1 Due to (6.30), we can represent ˜ ˇ ˜ ˚ f = (aF )+ f = ∇· E ∇(aF )– γ  , (6.33) ∗  1 1∗  1 ∗ s–2 where f ∈ H ,which,due to,e.g., [31, Theorem 2.10], can be in turn represented as 1∗ ∗ s– f =–γ  with some  ∈ H (∂). Then (6.30)issatisfied, and 1∗ ∗ ∗ + –1 ˜ ˜ ˇ F = T (f , aF )= γ f – (aF ) 2 ∗ 1 ∗ 1 ∗ ∗ –1 –1 ∗ = γ f =– γ γ  =– , (6.34) 1∗ ∗ ∗ –1 ∗ ∗ ∗ –1 –s because (γ ) γ  , w = γ  , γ w =  , w for any w ∈ H (∂). Hence ∗ ∂ ∗  ∗ ∂ (6.33)reduces to (6.26). Now (6.28)can be writteninthe form W (a )= F in , (6.35) where ˜ ˇ F := –aF + P f =–aF + P (aF )+ γ F (6.36) ∗ 1 1 2 is a harmonic function in  due to (6.29). The trace of equation (6.35)gives – a + W (a )= γ F on ∂. (6.37) 1 1 1 s– s– 2 2 Since the operator – I + W : H (∂) → H (∂) is an isomorphism (see Theo- rem 7.3), this implies –1 1 1 = – I + W γ F a 2 –1 1 1 + ∗ = – I + W γ –aF + P (aF )+ γ F , 1 2 a 2 which coincides with (6.27). Relations (6.26)and (6.27)can be writtenas (f ,  )= C (F , F ), where C : H () × ∗ ∗ ∗ 1 2 ∗ 3 1 s– s–2 s– 2 2 H (∂) → H () ×H (∂) is a linear continuous operator, as claimed. We still have to check that the functions f and  ,given by (6.26)and (6.27), satisfy equations (6.24) ∗ ∗ and (6.25). Indeed,  given by (6.27) satisfies equation (6.37)with F given by (6.36), and + + thus γ W (a )= γ F .Since both W (a )and F are harmonic functions belonging Mikhailov Boundary Value Problems (2018) 2018:87 Page 38 of 52 to the space H (), this implies (6.35)and,by (6.26), also (6.24). Finally, (6.26)implies by (6.34)that(6.32) is satisfied, and adding (6.31)toitleads to (6.25). s–2 Let us now prove that the operator C is unique. Indeed, let a couple (f ,  ) ∈ H () × ∗ ∗ ∗ s– H (∂) be a solution of linear system (6.24)–(6.25)with F =0 and F =0. Then (6.30) 1 2 s–2 s–2 + ˜ ˜ ˜ implies that r f =0 in , i.e., f ∈ H ⊂ H (). Hence, (6.32)reduces to 0 = T (f ,0) ∗ ∗ ∗ on ∂. By the first Green identity (2.15)thisgives + + 2–s ˜ ˜ 0= T (f ,0), γ v = f , v ∀ v ∈ H (), ∗ ∗ which implies f =0 in R . Finally, (6.27)gives  = 0. Hence, any solution of non- ∗ ∗ homogeneous linear system (6.24)–(6.25) has only one solution, which implies the unique- ness of the operator C . 1 3 Theorem 6.7 Let  be a bounded simply connected Lipschitz domain, < s < , and a ∈ 2 2 C () with σ = max{1, s}. The co-kernel of operator (6.14) is spanned over the functional ∗1 g := (0, 1) (6.38) 3 3 s s– ∗ –s –s ∗1 + 0 2 2 in [H () × H (∂)] = H () × H (∂), i.e., g (F , F )= F , γ u , where 1 2 2 ∂ u =1. Proof Let us consider the equation N U =(F , F ) , i.e., the BDIE system (N1 )for 1 2 (u, ϕ) ∈ H () × H (∂), u + Ru + W ϕ = F in , (6.39) + ∇ T A u; aRu + L (aϕ)= F on ∂, (6.40) s s– with arbitrary (F , F ) ∈ H () × H (∂). By Lemma 6.6 the right-hand side of the sys- 1 2 tem can be presented in the form (6.24)–(6.25), i.e., system (6.39)–(6.40)reduces to u + Ru + W(ϕ +  )= Pf in , (6.41) ∗ ∗ + ∇ + ˜ ˜ T A u; aRu + L (aϕ + a )= T (f ; Pf )on ∂, (6.42) ∗ ∗ ∗ s–2 s– where the couple (f ,  ) ∈ H () × H (∂)is given by (6.26)–(6.27). Up to the no- ∗ ∗ tations, system (6.41)–(6.42)isthe same as (6.4)–(6.5) with the right-hand side given by (6.6), where ψ =0. First, let s = 1. Then Theorems 6.1 and 6.3 imply that BDIE system (6.41)–(6.42)and hence (6.39)–(6.40) are solvable if and only if 0 ∗ 0 ˜ ˇ f , u = (aF )+ γ F , u ∗  1 2 ∗ 0 = ∇· E ∇(aF )+ γ F , u 1 2 n 0 + 0 =– E ∇(aF ), ∇u + F , γ u 1 2 R ∂ + 0 = F , γ u = 0, (6.43) ∂ Mikhailov Boundary Value Problems (2018) 2018:87 Page 39 of 52 0 n ∗1 where we took into account that u =1 in R . Thus the functional g defined by (6.38) generates the necessary and sufficient solvability condition of equation N U =(F , F ) . 1 2 ∗1 Hence g is a basis of the co-kernel of N (and thus the kernel of the operator N adjoint to N )for s =1. 1 3 Let us now choose any s ∈ ( , ). By Theorem 6.4,operator(6.14) and thus its adjoint 2 2 are Fredholm with zero index. We already proved that, at s = 1, the kernel of the adjoint ∗1 operator is spanned over g . For any fixed coefficient a ∈ C (), the operator 1   3 s s – s s – 2 2 N : H () × H (∂) → H () × H (∂) (6.44) is continuous for any s ∈ ( , σ]and particularly for s = s and s = 1. Then Lemma 7.5 implies that the co-kernel of operator (6.44)isthe same for s = s and s = 1 and is spanned ∗1 over g . 1 3 Lemma 6.8 Let  be a bounded simply connected Lipschitz domain, < s < , and 2 2 2 s s– a ∈ C (). For any couple (F , F ) ∈ H () × H (∂), there exists a unique couple + 1 2 s– s–2 (f ,  ) ∈ H () × H (∂) such that ∗∗ ∗∗ F = Pf – W  in , (6.45) 1 ∗∗ ∗∗ + + ˜ ˜ ˜ F = T (f + E R f ; Pf )– T W  on ∂. (6.46) 2 ∗∗  ∗ ∗∗ ∗∗ ∗∗ 3 1 s s– s–2 s– 2 2 Moreover,(f ,  )= C (F , F ), and C : H () × H (∂) → H () × H (∂) is ∗∗ ∗∗ ∗∗ 1 2 ∗∗ a linear continuous operator given by ∗ + f = (aF )+ γ F + γ F ∂ a , (6.47) ∗∗  1 2 1 n –1 1 1 + ∗ + = – I + W γ –aF + P (aF )+ γ F + γ F ∂ a . (6.48) ∗∗ 1 2 1 n a 2 s–2 s– Proof Let us first assume that there exist (f ,  ) ∈ H () × H (∂)satisfying equa- ∗∗ ∗∗ tions (6.45)and (6.46) and prove that they are then expressed in terms of F and F by 1 2 (6.47)–(6.48). Let us rewrite (6.45)as F – Pf =–W  in , (6.49) 1 ∗∗ ∗∗ Multiplying (6.49)by a and applying the Laplace operator to it, we obtain ˜ ˜ (aF – P f )= (aF )– f =– W (a )=0 in , (6.50) ∗∗ 1 ∗∗ ∗∗ which means that (aF )= r f in  (6.51) 1  ∗∗ s,0 and aF – P f ∈ H (; ). By equality (6.49) and the continuity of operator (3.73)in ∗∗ 1,0 Theorem 3.5,wealsohave F – Pf ∈ H (; A), which implies that the canonical co- 1 ∗∗ Mikhailov Boundary Value Problems (2018) 2018:87 Page 40 of 52 normal derivative T (F – Pf ) is well defined. Applying the canonical co-normal deriva- 1 ∗∗ tive operator T to both sides of equation (6.49), we obtain + + + ˜ ˜ ˜ ˜ –T W  = T (F – Pf )= T A(F – Pf ); F – Pf ∗∗ 1 ∗∗ 1 ∗∗ 1 ∗∗ ˚ ˜ ˜ = T E ∇· a∇(F – Pf ) ; F – Pf 1 ∗∗ 1 ∗∗ ˚ ˜ ˚ ˜ ˜ = T E (aF – P f )– E ∇· (F – Pf )∇a ; F – Pf ∗∗  1 ∗∗ 1 ∗∗ ˚ ˚ ˜ ˜ = T –E ∇· (F ∇a)– E R f ; F – Pf , (6.52) 1  ∗ ∗∗ 1 ∗∗ where (6.50) and the third relation in (3.16) were taken into account. Substituting this into (6.46), we obtain ˜ ˚ F = T f – E ∇· (F ∇a), F on ∂. (6.53) 2 ∗∗  1 1 Due to (6.51), we can represent ˜ ˇ ˜ ˚ f = (aF )+ f = ∇· E ∇(aF )– γ  , (6.54) ∗∗  1 1∗  1 ∗∗ s–2 where f ∈ H ,which,due to,e.g., [31, Theorem 2.10], can be in turn represented as 1∗ ∗ s– f =–γ  with some  ∈ H (∂). Then (6.51)issatisfied, and 1∗ ∗∗ ∗∗ + –1 ˜ ˚ ˜ ˚ ˇ F = T f – E ∇· (F ∇a), F = γ f – E ∇· (F ∇a)– AF 2 ∗∗  1 1 ∗∗  1 1 –1 ∗ ˚ ˚ ˚ = γ ∇· E ∇(aF )– γ  – E ∇· (F ∇a)– ∇· E (a∇F ) 1 ∗∗  1  1 –1 ∗ + ˚ ˚ = γ ∇· E (F ∇a)– γ  – E ∇· (F ∇a) =– – γ F ∂ a, (6.55) 1 ∗∗  1 ∗∗ 1 n –s because for any w ∈ H (∂), –1 ∗ ˚ ˚ γ ∇· E (F ∇a)– γ  – E ∇· (F ∇a) , w 1 ∗∗  1 ∗ –1 ˚ ˚ = ∇· E (F ∇a)– γ  – E ∇· (F ∇a), γ w 1 ∗∗  1 –1 ∗ –1 –1 ˚ ˚ = ∇· E (F ∇a), γ w – γ  , γ w – E ∇· (F ∇a), γ w 1 n ∗∗  1 –1 –1 =– E (F ∇a), ∇γ w –  , w + F ∇a, ∇γ w 1 n ∗∗ ∂ 1 + + –1 + – n · γ (F ∇a), γ γ w =– γ F ∂ a, w –  , w . 1 1 n ∗∗ ∂ ∂ ∂ Hence (6.53)reduces to  =–F –(γ F )∂ a,and (6.54)to(6.47). ∗∗ 2 1 n Now (6.49)can be writteninthe form W (a )= F in , (6.56) ∗∗ where ∗ + ˜ ˇ F := –aF + P f =–aF + P (aF )+ γ F + γ F ∂ a (6.57) ∗∗ 1 1 2 1 n is a harmonic function in  due to (6.50). The trace of equation (6.56)gives – a + W (a )= γ F on ∂. (6.58) ∗∗ ∗∗ 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 41 of 52 1 1 1 s– s– 2 2 Since the operator – I + W : H (∂) → H (∂) is an isomorphism (see Theo- rem 7.3), this implies –1 1 1 = – I + W γ F ∗∗ a 2 –1 1 1 + ∗ + = – I + W γ –aF + P (aF )+ γ F + γ F ∂ a , 1 2 1 n a 2 which coincides with (6.48). Relations (6.47)and (6.48)can be writtenas (f ,  )= C (F , F ), where C : H () × ∗∗ ∗∗ ∗∗ 1 2 ∗∗ 3 1 s– s–2 s– 2  2 H (∂) → H () × H (∂) is a linear continuous operator, as claimed. We still have to check that the functions f and  given by (6.47)and (6.48)satisfy equa- ∗∗ ∗∗ tions (6.45)and (6.46). Indeed,  given by (6.48) satisfies equation (6.58), and thus ∗∗ + + γ W (a )= γ F .Since both W (a )and F are harmonic functions belonging ∗∗ ∗∗ to the space H (), this implies (6.56)–(6.57)and by (6.47)also(6.45). Finally, (6.47)im- plies by (6.55)that(6.53) is satisfied, and adding (6.52)toitleads to (6.46). Let us now prove that the operator C is unique. Indeed, let a couple (f ,  ) ∈ ∗∗ ∗∗ ∗∗ s– s–2 H () × H (∂) be a solution of linear system (6.45)–(6.46)with F =0 and F =0. 1 2 s–2 s–2 ˜ ˜ Then (6.51)implies that r f =0 in , i.e., f ∈ H ⊂ H (). Hence, (6.53)reduces to ∗∗ ∗∗ 0= T (f ,0) on ∂. By the first Green identity (2.15)thisgives ∗∗ + + 2–s ˜ ˜ 0= T (f ,0), γ v = f , v ∀ v ∈ H (), ∗∗ ∗∗ which implies f =0 in R . Finally, (6.48)gives  = 0. Hence, non-homogeneous linear ∗∗ ∗∗ system (6.45)–(6.46) has only one solution, which implies the uniqueness of the opera- tor C . ∗∗ 1 3 Theorem 6.9 Let  be a bounded simply connected Lipschitz domain, < s < , and a ∈ 2 2 C (). The co-kernel of operator (6.15) is spanned over the functional ∗1 + g := γ ∂ a,1 (6.59) 3 3 s s– ∗ –s –s 2 2 in [H () × H (∂)] = H () × H (∂), i.e., ∗1 + + 0 g (F , F )= γ F ∂ a + F , γ u , 1 2 1 n 2 where u =1. Proof Let us consider the equation N U =(F , F ) , i.e., the BDIE system (N1) for (u, ϕ) ∈ 1 2 s s– H () × H (∂), u + Ru + W ϕ = F in , (6.60) + + + T Ru + T W ϕ = F on ∂, (6.61) 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 42 of 52 s s– with arbitrary (F , F ) ∈ H () × H (∂). By Lemma 6.8 the right-hand side of the sys- 1 2 tem has form (6.45)–(6.46), i.e., system (6.60)–(6.61)reduces to u + Ru + W(ϕ +  )= Pf in , (6.62) ∗∗ ∗∗ + + + ˜ ˚ ˜ ˜ T Ru + T W(ϕ +  )= T (f + E R f , Pf )on ∂, (6.63) ∗∗ ∗∗  ∗ ∗∗ ∗∗ s–2 s– where the couple (f ,  ) ∈ H () × H (∂)is given by (6.47)–(6.48). Up to the no- ∗∗ ∗∗ tations, system (6.62)–(6.63)isthe same as (6.7)–(6.8) with the right-hand side given by (6.9), where ψ =0. First, let s = 1. Then Theorems 6.1 and 6.3 imply that BDIE system (6.62)–(6.63)and hence (6.60)–(6.61) are solvable if and only if 0 ∗ + 0 f , u = (aF )+ γ F + γ F ∂ a , u ∗∗  1 2 1 n ∗ + 0 = ∇· E ∇(aF )+ γ F + γ F ∂ a , u 1 2 1 n 0 + + 0 =– E ∇(aF ), ∇u + F + γ F ∂ a, γ u 1 n 2 1 n R ∂ + + 0 = γ F ∂ a + F , γ u =0, 1 n 2 0 n ∗1 wherewetookintoaccountthat u =1 in R . Thus the functional g defined by (6.59) generates the necessary and sufficient solvability condition of equation N U =(F , F ) . 1 2 ∗1 1 Hence g is a basis of the co-kernel of N (and thus the kernel of the operator adjoint to N )for s =1. 1 3 Let now s ∈ ( , ). By Theorem 6.4 operator (6.15) and thus its adjoint are Fredholm with 2 2 zero index. We already proved that, at s = 1, the kernel of the adjoint operator is spanned ∗1 1 3 over g . Then Lemma 7.5 implies that the kernel is the same for any s ∈ ( , ). 2 2 To find the co-kernel of operator (6.16), we need some auxiliary assertions. Lemma 6.10 and Theorem 6.11 were proved in [33, Lemma 6.4 and Theorem 6.5] for the infinitely smooth coefficient a and boundary ∂. We further only slightly modify these proofs for the non-smooth coefficients and Lipschitz boundary. Lemma 6.10 Let  be a bounded simply connected Lipschitz domain, s > , a ∈ C (), s–2 and f ∈ H (). If r Pf =0 in , (6.64) then f =0 in R . Proof Multiplying (6.64)by a, taking into account (3.16), and applying the Laplace oper- s–2 3 ˜ ˜ ˜ ator, we obtain r f =0, which means f ∈ H .If s ≥ ,then f = 0 by Theorem 2.10 from 1 3 s– ∗ [31]. If < s < , then by the same theorem there exists v ∈ H (∂)such that f = γ v. 2 2 ∗ n This gives Pf = Pγ v =–Vv in R ;see (3.53). Then (6.64)reduces to Vv =0 in ,which implies v =0 on ∂ by Lemma 4.8(i), and thus f =0 in R .  Mikhailov Boundary Value Problems (2018) 2018:87 Page 43 of 52 1 3 Theorem 6.11 Let  be a bounded simply connected Lipschitz domain, < s < , and 2 2 a ∈ C (). The operator s–2 s r P : H () → H () (6.65) and its inverse –1 s s–2 (r P) : H () → H () (6.66) are continuous, and –1 –1 + ∗ –1 + n s (r P) g = E I – r V V γ – γ V γ (ag) in R , ∀ g ∈ H (). (6.67) Proof The continuity of (6.65)isgiven by Theorem 3.2. By Lemma 6.10 operator (6.65)is injective. Let us prove its surjectivity. To this end, for arbitrary g ∈ H (), let us consider s–2 the following equation with respect to f ∈ H (): r P f = g in . (6.68) Let g ∈ H () be the (unique) solution of the following Dirichlet problem: g =0 in , 1 1 + + –1 + γ g = γ g, which can be particularly presented as g = V V γ g; see, e.g., [11]orproof 1 1 s + of Lemma 2.6 in [31]. Let g := g – r g .Then g ∈ H ()and γ g =0, and thus g can be 0  1 0 0 0 uniquely extended to E g ∈ H (). Thus by (3.53)equation(6.68)takes form ∗ –1 + r P f + γ V γ g = g in . (6.69) s–2 n Any solution f ∈ H () of the corresponding equation in R , ∗ –1 + n ˜ ˚ P f + γ V γ g = E g in R , (6.70) evidently solves (6.69). If f solves (6.70), then applying the Laplace operator to (6.70), we obtain ∗ –1 + –1 + ∗ –1 + n ˜ ˜ ˚ ˚ f = Qg := E g – γ V γ g = E g – r V V γ g – γ V γ g in R . (6.71) On the other hand, substituting f given by (6.71)into(6.70) and taking into account that ˜ ˜ ˜  ˜ h = h for any h ∈ H (), s ∈ R,weobtainthat Qg is indeed a solution of equation (6.70)and thus of (6.69). By Lemma 6.10 the solution of (6.69) is unique, which means –1 ˜ ˜ that the operator Q is inverse to operator (6.65), i.e., Q =(r P) .Since is a continuous s s–2 –1 operator from H ()to H (), equation (6.71) implies that the operator (r P) = Q : s s–2 1 H () → H () is continuous. The relations P = P and a(x) ≥ a >0 then imply the min invertibility of operator (6.65) and ansatz (6.67). 1 3 Theorem 6.12 Let  be a bounded simply connected Lipschitz domain, < s < , and a ∈ 2 2 C () with σ = max{1, s}. The co-kernel of operator (6.16) is spanned over the functional +∗  –1 + 0 –aγ ( + W )V γ u ∗2 g := (6.72) –1 + 0 –a( – W )V γ u 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 44 of 52 1 1 s s– ∗ –s –s 2  2 in [H () × H (∂)] = H () × H (∂), i.e., ∗2 g (F , F ) 1 2 1 1 +∗  –1 + 0  –1 + 0 = –aγ + W V γ u , F + –a – W V γ u , F , 1 2 2 2 where u (x)=1. Proof Let us consider the equation N U =(F , F ) , i.e., the BDIE system (N2), 1 2 u + Ru + W ϕ = F in , (6.73) ϕ + γ Ru + W ϕ = F on ∂, (6.74) 1 1 s s– s s– 2 2 with arbitrary (F , F ) ∈ H () × H (∂)for (u, ϕ) ∈ H () × H (∂). 1 2 Introducing the new variable ϕ = ϕ –(F – γ F ), BDIE system (6.73)–(6.74)takes form 2 1 u + Ru + W ϕ = F in , (6.75) +  + ϕ + γ Ru + W ϕ = γ F on ∂, (6.76) where + s F = F – W F – γ F ∈ H (). 1 2 1 On the other hand, by Theorem 6.11, we can always represent F = Pf ,with –1 + +∗ –1 +  s–2 f = E I – r V V γ – γ V γ aF ∈ H (). For F = Pf , the right-hand side of BDIE system (6.73)–(6.74)isthe same as in (6.12)with ˜ ˜ f = f and ψ =0. ∗ 0 First, let s = 1. Then Theorems 6.1 and 6.3 imply that BDIE system (6.75)–(6.76)issolv- able if and only if 0 –1 + +∗ –1 +  0 ˜ ˚ f , u = E I – r V V γ – γ V γ aF , u 1 n –1 +  0 –1 +  + 0 = E I – r V V γ aF , u – V γ aF , γ u 1 n R ∂ +  –1 + 0 =– γ aF , V γ u + + –1 + 0 =– γ (aF )+(aF ) – W a F – γ F , V γ u 1 2 2 1 1 1 +∗  –1 + 0  –1 + 0 =– F , aγ + W V γ u – F , a – W V γ u 1 2 2 2 = 0. (6.77) ∗2 Thus the functional g defined by (6.72) generates a necessary and sufficient solvability 2  ∗2 condition of equation N U =(F , F ) .Hence g is a basis of the co-kernel of operator 1 2 (6.16)for s =1. Mikhailov Boundary Value Problems (2018) 2018:87 Page 45 of 52 1 3 Let us now choose any s ∈ ( , ). By Theorem 6.4 operator (6.16) and thus its adjoint 2 2 are Fredholm with zero index. We already proved that, at s = 1, the kernel of the adjoint ∗2 σ operator is spanned over g . For any fixed coefficient a ∈ C (), the operator 1   1 2 s s – s s – 2 2 N : H () × H (∂) → H () × H (∂) (6.78) is continuous for any s ∈ ( , σ]and particularly for s = s and s = 1. Then Lemma 7.5 implies that the co-kernel of operator (6.78)isthe same for s = s and s = 1 and is spanned ∗2 over g . Theorems 6.3, 6.5,and 6.7 (or 6.9) imply the following extension of Theorem 6.1 to the 1 3 range < s < . 2 2 1 3 Corollary 6.13 Let  be a bounded simply connected Lipschitz domain, < s < , f ∈ 2 2 s– s–2 σ H (), ψ ∈ H (∂), and a ∈ C () with σ = max{1, s}. The homogeneous Neumann problem (6.1)–(6.2) admits only one linearly independent 0 s solution u =1 in H (). The non-homogeneous Neumann problem (6.1)–(6.2) is solvable in H () if and only if condition (6.3) is satisfied. Proof Assuming that a function u is a solution of the homogeneous Neumann problem, by Theorem 6.3 the couple (u, ϕ)=(u, γ ϕ) solves the homogeneous BDIE system (N1 ), and then Theorem 6.7 implies that u is spanned over u =1. Assume that solvability condition (6.3) is satisfied. Then the right-hand side (6.6)ofthe ∗1 + 0 BDIE system (N1 ) satisfies its solvability condition g (F , F )= F , γ u =0 given 1 2 2 ∂ by Theorem 6.7. Indeed, due to the first Green identities (2.15)and (2.18) applied to the operator and Remark 2.7,since V ψ is a harmonic function in  and u =1, we obtain + 0 +  + 0 ˜ ˜ F , γ u = T (f ; P f )– ψ + W ψ , γ u 0 0 + + + 0 ˜ ˜ = T (f ; P f )– ψ + T V ψ , γ u 0 0 + 0 ˜ ˇ ˜ = f , u + E P f , u – ψ , γ u 0 0 ˜ ˇ V ψ , u + E V ψ , u 0 + 0 = f , u – ψ , γ u . (6.79) Hence the BDIE system (N1 ) is solvable, implying solvability of the Neumann BVP due to Theorem 6.3(ii). This proves that condition (6.3)issufficient. Let us now assume that there exists a solution of the Neumann BVP. Hence Theo- rem 6.3(i) implies that the BDIE system (N1 ) with the right-hand side (6.9)issolvable, + 0 implying that its solvability condition F , γ u =0 is satisfied. Then (6.79)implies 2 ∂ condition (6.3), proving that it is necessary. 6.3 Perturbed (stabilised) segregated BDIE systems for the Neumann problem Theorem 6.5 implies that even when the solvability condition (6.3)issatisfied, the solutions of BDIE systems (N1 ), (N1), and (N2) are not unique, and moreover, the 1 2 BDIE left-hand side operators N , N ,and N , have non-zero kernels and thus are Mikhailov Boundary Value Problems (2018) 2018:87 Page 46 of 52 not invertible. To find a solution (u, ϕ) from uniquely solvable BDIE systems with continuously invertible left-hand side operators, let us consider, following [28], some stabilised BDIE systems obtained from (N1 ), (N1), and (N2) by finite-dimensional operator perturbations. Note that other choices of the perturbing operators are also pos- sible. N  0 We further use the notations U =(u, ϕ) , U = (1, 1) ,and |∂| := dS. Let us introduce the perturbed counterparts of the BDIE systems (N1 ), (N1), and (N2): N N1 1 N N1 2 N N2 ˆ ˆ ˆ N U = F , N U = F , N U = F , (6.80) 1 1 1 2 2 2 ˆ ˚ ˆ ˚ ˆ ˚ where N := N + N , N := N + N , N := N + N ,and 1 0 N 1 N 0 N 1 ˚ ˚ N U (y)= N U (y):= g U G (y)= ϕ(x) dS , (6.81) |∂| 1 that is, 1 0 0 N 1 g U := ϕ(x) dS, G (y):= , (6.82) |∂| 1 whereas –1 1 a (y) 2 N 0 N 2 N U := g U G = ϕ(x) dS , + –1 |∂| γ a (y) 0 N that is, g (U ) isasin(6.82), and –1 a (y) G (y):= . + –1 γ a (y) 1 3 Theorem 6.14 Let  be a bounded simply connected Lipschitz domain, < s < , and 2 2 σ = max{1, s}. (i) The following operators are continuous and continuously invertible: 1 3 s s– s s– σ 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (), (6.83) 1 3 1 s s– s s– 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (), (6.84) 1 1 2 s s– s s– σ 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (). (6.85) ∗1 N1 ∗1 N1 ∗2 N2 (ii) If the conditions g (F )=0, g (F )=0, or g (F )=0 are satisfied, then the unique solutions of the perturbed BDIE systems in (6.80) give the solutions U of the corresponding original BDIE systems (N1 ), (N1), and (N2) such that 1 1 0 N + g U = ϕ dS = γ udS =0. |∂| |∂| ∂ ∂ Mikhailov Boundary Value Problems (2018) 2018:87 Page 47 of 52 ∗1 ∗1 Proof For the functional g given by (6.38)inTheorem 6.7, g (G )= |∂|. Similarly, ∗1 ∗1 1 for the functional g given by (6.59)inTheorem 6.9, g (G )= |∂|. ∗2 –1 For the functional g given by (6.72)inTheorem 6.12, since the operator V : 1 1 2 2 H (∂) → H (∂) is positive definite and u (x) = 1, there exists a positive constant C such that ∗2 2 +∗  –1 + 0 –1 0 g G = –aγ + W V γ u , a u –1 + 0 + –1 0 + –a – W V γ u , γ a u 1 1 –1 + 0  –1 + 0 + 0 =– + W V γ u + – W V γ u , γ u 2 2 –1 + 0 + 0 =– V γ u , γ u 2 2 + 0 + 0 ≤ –C γ u 1 ≤ –C γ u L (∂) 2 2 H (∂) =–C|∂| < 0. (6.86) 0 0 On the other hand, g (U )=1. Hence Theorem 7.4 from [28]implies theclaimsofthe theorem. 7 Auxiliary assertions We provide here some auxiliary results used in the main text. 1 3 s σ Theorem 7.1 Let < s < , u ∈ H (), a ∈ C () with σ = max{1, s}, Au = r finaninte- 2 2 s–2 2 rior or exterior Lipschitz domain  for some f ∈ H (). Let {f }∈ H () be a sequence ˜ ˚ such that f – E f → 0 as k →∞. s–2 H () s,– Then there exists a sequence {u }∈ H (; A) such that Au = f in  and u – k k k + + u → 0 as k →∞. Moreover, T (u )– T (f ; u) 3 → 0 as k →∞. k H () k s– H (∂) Proof Let us consider the Dirichlet problem Au = f in , (7.1) k k + + γ u = γ u on ∂, (7.2) s D –1 By Corollary 5.5theuniquesolutionofproblem(7.1)–(7.2)in H ()is u =(A ) (f , ϕ ) , k k k D –1 s–2 s– s where (A ) : H () × H (∂) → H () is a continuous operator. Hence the func- s 2 tions u converge to u in H ()as k →∞.Since Au = f ∈ H (), we obtain that k k k • s,– + u ∈ H (; A) and the canonical conormal derivative T u is well defined. Then sub- k k tracting (2.16)for u from (2.12), we obtain + + –1 ˜ ˜ ˇ T (f ; u)– T u = γ f – E f + A (u – u ) . k  k  k Hence + + ˜ ˜ ˚ T (f ; u)– T u 3 ≤ C f – E f s–2 + C u – u (7.3) k  k  1 k H () s– H () H (∂) Mikhailov Boundary Value Problems (2018) 2018:87 Page 48 of 52 for some positive C and C . Since the right-hand side of (7.3)tends to zero as k →∞,so does the left-hand side. 1 1 – – 2 s–2 2 Note that since D() ⊂ H ()is dense in H (), the sequence {f }∈ H ()from • • the hypotheses of Theorem 7.1 does always exist. The following multiplication theorem is well known; see, e.g., [15, Theorems 1.4.1.1, 1.4.1.2], [54, Theorem 2(b)], [1, Theorems 1.9.1, 1.9.2, 1.9.5], [32, Theorem 3.2]. Theorem 7.2 Let  be an open set. (i) If g ∈ L ( ), then gv ∈ L ( ) and gv ≤ c g v for every ∞ 0 2 0 L () L ( ) L ( ) 2 0 2 0 v ∈ L ( ). 2 0 |σ |–1,1 σ (ii) If σ is a non-zero integer and g ∈ C ( ), then gv ∈ H ( ) for every 0 0 v ∈ H ( ), and gv σ ≤ c g v σ . 0 H () |σ |–1,1 H ( ) C ( ) 0 (iii) If σ is a non-integer, |σ | = m + θ, where m is a non-negative integer and 0< θ <1, m,η σ then for g ∈ C ( ) with θ < η <1, we have gv ∈ H ( ) and 0 0 gv σ ≤ c g v σ for every v ∈ H ( ). m,η H () H ( ) 0 C ( ) 0 In all cases, c is a positive constant independent of g, v, or  . Theorem 7.3 Let  be a bounded simply connected Lipschitz domain, and let 0 ≤ σ ≤ 1. The operators σ –1 σ V : H (∂) → H (∂), (7.4) σ σ – I + W : H (∂) → H (∂), (7.5) –σ –σ – I + W : H (∂) → H (∂) (7.6) are isomorphisms, and the operators σ σ I + W : H (∂) → H (∂), (7.7) –σ –σ I + W : H (∂) → H (∂), (7.8) σ σ –1 L : H (∂) → H (∂) (7.9) are Fredholm with zero index. Proof The properties of the boundary integral operators (7.4)–(7.9) related to the har- monic layer potential are well known; see, e.g., [51], [39, Theorem 4.1], [14, Theorem 8.1] for the invertibility of operators (7.4)–(7.6) and the Fredholm properties of operators (7.7)–(7.8). The Fredholm property of operator (7.9)for σ = is also well known; see, e.g., [27, Theorem 7.8]. Then the corresponding result for 0 ≤ σ ≤ 1 can be proved as in [27, Theorem 7.17] by using a sharper regularity result from [11, Theorem 3]. Theorem 7.4 further is implied by [28, Lemma 2] (see also [50,Sect. 21],[49, Sect. 21.4], ∗ ∗ where the particular case h (x )= x ˚ (h )= δ has been considered). Another approach, j j ij i i although with hypotheses similar to those in Theorem 7.4,ispresented in [17, Lemma 4.8.24]. Mikhailov Boundary Value Problems (2018) 2018:87 Page 49 of 52 Theorem 7.4 Let B and B be two Banach spaces. Let A : B → B be a linear Fredholm 1 2 1 2 ∗ ∗ ∗ operator with zero index, and let A : B → B be the adjoint operator with dim ker A = 2 1 ∗ n ∗ ∗ n ∗ dim ker A = n < ∞, where ker A = span{x ˚ } ⊂ B and ker A = span{x ˚ } ⊂ B . Let i 1 i=1 i i=1 2 A x := h h (x), 1 i i=1 ∗ ∗ where h and h (i = 1,..., n) are elements from B and B , respectively, such that i 2 i 1 ∗ ∗ det h (x ˚ ) =0, det x ˚ (h ) =0, i, j = 1,..., n. (7.10) j j i i Then: (i) the operator A – A : B → B is an isomorphism; 1 2 (ii) if y ∈ B satisfies the solvability conditions x ˚ (y)=0, i = 1,..., n, (7.11) of the equation Ax = y, (7.12) then the unique solution x of equation (A – A )x = y, (7.13) is a solution of equation (7.12) such that h (x)=0 (i = 1,..., k). (7.14) (iii) Vice versa, if x is a solution of equation (7.13) satisfying conditions (7.14), then conditions (7.11) are satisfied for the right-hand side y of equation (7.13), and x is a solution of equation (7.12) with the same right-hand side y. Note that more results about finite-dimensional operator perturbations are available in [28]. The following known result (see, e.g., [42, Lemma 11.9.21]) is useful for us. Lemma 7.5 Let X , X and Y , Y , be Banach spaces such that the embeddings X → X 1 2 1 2 1 2 and Y → Y are continuous, and the embedding Y → Y has a dense range. Assume that 1 2 1 2 T : X → Y and T : X → Y are Fredholm operators with the same index, ind(T : X → 1 1 2 2 1 Y )= ind(T : X → Y ). Then Ker{T : X → Y } = Ker{T : X → Y }. 1 2 2 1 1 2 2 8 Concluding remarks The Dirichlet and Neumann problems on a bounded Lipschitz domain for a variable- s–2 coefficient second-order PDE with general right-hand side functions from H ()and 1 3 s–2 H (), < s < , respectively, were equivalently reduced to three direct segregated 2 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 50 of 52 boundary-domain integral equation systems for each of the BVPs. This involved system- atic use of the generalised co-normal derivatives. The operators associated with the left- hand sides of all the BDIE systems were analysed in the corresponding Sobolev spaces. It was shown that the operators of the BDIE systems for the Dirichlet problem are con- tinuous and continuously invertible. For the Neumann problem, the BDIE system opera- tors are continuous but only Fredholm with zero index; their kernels and co-kernels were analysed, and appropriate finite-dimensional perturbations were constructed to make the perturbed (stable) operators invertible and provide a solution of the original BDIE systems and the Neumann problem. The same approach can be implemented to extend to the general PDE right-hand sides, non-smooth coefficients and Lipschitz domains: the BDIE systems for the mixed prob- lems, unbounded domains, BDIEs of more general scalar PDEs and the systems of PDEs, and the united and localised BDIEs, for which the analysis is now available for the right- hand sides only from L (), with smooth coefficients and smooth domain boundaries; see [2–10, 13, 30, 36, 37]. Acknowledgements Not applicable. Funding This research was supported by the grants EP/H020497/1: “Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems” and EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK. Abbreviations BDIE, Boundary-domain integral equation; BVP, Boundary value problem; PDE, Partial differential equation. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Competing interests The author declares that he has no competing interests. Authors’ contributions The paper was written by the author personally. The author read and approved the final manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 12 January 2018 Accepted: 30 April 2018 References 1. Agranovich, M.S.: Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains. Springer, Cham (2015) 2. Ayele, T.G., Mikhailov, S.E.: Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP. Eurasian Math. J. 2(3), 20–41 (2011) 3. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. J. Integral Equ. Appl. 21(4), 499–543 (2009) 4. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of some localized boundary-domain integral equations. J. Integral Equ. Appl. 21(3), 405–445 (2009) 5. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics. J. Integral Equ. Appl. 22(1), 19–37 (2010) 6. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks. Numer. Methods Partial Differ. Equ. 27(1), 121–140 (2011). https://doi.org/10.1002/num.20639 7. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack. Mem. Differ. Equ. Math. Phys. 52, 17–64 (2011) 8. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains. Anal. Appl. 11(4), 1350006 (2013). https://doi.org/10.1142/S0219530513500061 Mikhailov Boundary Value Problems (2018) 2018:87 Page 51 of 52 9. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients. Integral Equ. Oper. Theory 76(4), 509–547 (2013). https://doi.org/10.1007/s00020-013-2054-4 10. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second order strongly elliptic PDE systems. Math. Methods Appl. Sci. 40, 1817–1837 (2017). https://doi.org/10.1002/mma.4100 11. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988) 12. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology Vol. 4: Integral Equations and Numerical Methods. Springer, Berlin (1990) 13. Dufera, T.T., Mikhailov, S.E.: Analysis of boundary-domain integral equations for variable-coefficient Dirichlet BVP in 2D. In: Constanda, C., Kirsh, A. (eds.) Integral Methods in Science and Engineering: Theoretical and Computational Advances. Springer, Boston (2015). https://doi.org/10.1007/978-3-319-16727-5_15 14. Fabes, E., Mendez, O., Mitrea, M.: Boundary layers on Sobolev–Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159, 323–368 (1998) 15. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985) 16. Grzhibovskis, R., Mikhailov, S., Rjasanow, S.: Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D. Comput. Mech. 51, 495–503 (2013). https://doi.org/10.1007/s00466-012-0777-8 17. Hackbusch, W.: Integral Equations: Theory and Numerical Treatment. International Series of Numerical Mathematics, vol. 120. Birkhäuser, Basel (1995) 18. Haroske, D., Triebel, H.: Distributions, Sobolev Spaces, Elliptic Equations. EMS Textbooks in Mathematics. Eur. Math. Soc., Zürich (2008) 19. Hellwig, G.: Partial Differential Equations: An Introduction. Teubner, Stuttgart (1977) 20. Hilbert, D.: Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen, 2nd edn. Teubner, Leipzig (1924) 21. Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, Springer, Berlin (2008) 22. Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull., New Ser., Am. Math. Soc. 4, 203–207 (1981) 23. Jerison, D.S., Kenig, C.E.: The Dirichlet problem in non-smooth domains. Ann. Math. 113, 367–382 (1981) 24. Jerison, D.S., Kenig, C.E.: Boundary value problems on Lipschitz domains. In: Littman, W. (ed.) Studies in Partial Differential Equations, pp. 1–68. Math. Assoc. of America, Washington (1982) 25. Levi, E.E.: I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali. Mem. Soc. Ital. dei Sc. XL 16, 1–112 (1909) 26. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972) 27. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000) 28. Mikhailov, S.E.: Finite-dimensional perturbations of linear operators and some applications to boundary integral equations. Eng. Anal. Bound. Elem. 23, 805–813 (1999) 29. Mikhailov, S.E.: Localized boundary-domain integral formulations for problems with variable coefficients. Eng. Anal. Bound. Elem. 26, 681–690 (2002) 30. Mikhailov, S.E.: Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficient. Math. Methods Appl. Sci. 29, 715–739 (2006) 31. Mikhailov, S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011). https://doi.org/10.1016/j.jmaa.2010.12.027 32. Mikhailov, S.E.: Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains. J. Math. Anal. Appl. 400(1), 48–67 (2013). https://doi.org/10.1016/j.jmaa.2012.10.045 33. Mikhailov, S.E.: Analysis of segregated boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems with general data. (2015) arXiv:1509.03501 34. Mikhailov, S.E., Mohamed, N.A.: Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient. Int. J. Comput. Math. 89(11), 1488–1503 (2012). https://doi.org/10.1080/00207160.2012.679733 35. Mikhailov, S.E., Nakhova, I.S.: Mesh-based numerical implementation of the localized boundary-domain integral equation method to a variable-coefficient Neumann problem. J. Eng. Math. 51, 251–259 (2005) 36. Mikhailov, S.E., Portillo, C.F.: BDIE system to the mixed BVP for the Stokes equations with variable viscosity. In: Constanda, C., Kirsh, A. (eds.) Integral Methods in Science and Engineering: Theoretical and Computational Advances. Springer, Boston (2015). https://doi.org/10.1007/978-3-319-16727-5_33 37. Mikhailov, S.E., Portillo, C.F.: A new family of boundary-domain integral equations for a mixed elliptic BVP with variable coefficient. In: Harris, P. (ed.) Proceedings of the 10th UK Conference on Boundary Integral Methods, pp. 76–84. University of Brighton, Brighton (2015) 38. Miranda, C.: Partial Differential Equations of Elliptic Type, 2nd edn. Springer, Berlin (1970) 39. Mitrea, D.: The method of layer potentials for non-smooth domain with arbitrary topology. Integral Equ. Oper. Theory 29, 320–338 (1997) 40. Mitrea, I., Mitrea, M.: Multy-Layer Potentials and Boundary Problems for Higher-Order Elliptic Systems in Lipschitz Domains. Lecture Notes in Mathematics, vol. 2063. Springer, Berlin (2013) 41. Mitrea, M., Monniaux, S.: The regularity of the Stokes operator and the Fujita–Kato approach to the Navier–Stokes initial value problem in Lipschitz domains. J. Funct. Anal. 254, 1522–1574 (2008) 42. Mitrea, M., Wright, M.: Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains. Astérisque, vol. 344. Société Mathématique de France, Paris (2012) 43. Pomp, A.: The Boundary-Domain Integral Method for Elliptic Systems. With Applications in Shells. Lecture Notes in Mathematics, vol. 1683. Springer, Berlin (1998) 44. Pomp, A.: Levi functions for linear elliptic systems with variable coefficients including shell equations. Comput. Mech. 22, 93–99 (1998) Mikhailov Boundary Value Problems (2018) 2018:87 Page 52 of 52 45. Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter, Berlin (1996) 46. Sladek, J., Sladek, V., Atluri, S.N.: Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties. Comput. Mech. 24, 456–462 (2000) 47. Sladek, J., Sladek, V., Zhang, J.-D.: Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients. J. Eng. Math. 51, 261–282 (2005) 48. Taigbenu, A.E.: The Green Element Method. Kluwer Academic, Boston (1999) 49. Trenogin, V.A.: Functional Analysis. Nauka, Moscow (1980) 50. Vainberg, M.M., Trenogin, V.A.: Theory of Branching of Solutions of Non-Linear Equations. Noordhoff, Leyden (1974) 51. Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984) 52. Zhu, T., Zhang, J.-D., Atluri, S.N.: A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput. Mech. 21, 223–235 (1998) 53. Zhu, T., Zhang, J.-D., Atluri, S.N.: A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems. Eng. Anal. Bound. Elem. 23, 375–389 (1999) 54. Zolesio, J.L.: Multiplication dans les espaces de Besov. Proc. R. Soc. Edinb. 78A, 113–117 (1977) http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Boundary Value Problems Springer Journals

Analysis of segregated boundary-domain integral equations for BVPs with non-smooth coefficients on Lipschitz domains

Free
52 pages

Loading next page...
 
/lp/springer_journal/analysis-of-segregated-boundary-domain-integral-equations-for-bvps-A0N12H2m4d
Publisher
Springer Journals
Copyright
Copyright © 2018 by The Author(s)
Subject
Mathematics; Difference and Functional Equations; Ordinary Differential Equations; Partial Differential Equations; Analysis; Approximations and Expansions; Mathematics, general
eISSN
1687-2770
D.O.I.
10.1186/s13661-018-0992-0
Publisher site
See Article on Publisher Site

Abstract

Department of Mathematics, Segregated direct boundary-domain integral equations (BDIEs) based on a Brunel University London, Uxbridge, UK parametrix and associated with the Dirichlet and Neumann boundary value problems for the linear stationary diffusion partial differential equation with a variable Hölder-continuous coefficients on Lipschitz domains are formulated. The PDE s–2 s–2 right-hand sides belong to the Sobolev (Bessel potential) space H ()or H (), 1 3 < s < , when neither strong classical nor weak canonical co-normal derivatives are 2 2 well defined. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm property and invertibility of the BDIE operators are analysed in appropriate Sobolev spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible; however, some finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators. MSC: 35J25; 31B10; 45K05; 45A05 Keywords: Partial differential equations; Non-smooth coefficients; Sobolev spaces; Parametrix; Integral equations; Equivalence; Lipschitz domain; Invertibility 1 Introduction Many applications in science and engineering can be modelled by boundary-value prob- lems (BVPs) for partial differential equations with variable coefficients. Reduction of the BVPs with arbitrarily variable coefficients to explicit boundary integral equations is usu- ally not possible, since the fundamental solution needed for such reduction is generally not available in an analytical form (except for some special dependence of the coefficients on coordinates). Using a parametrix (Levi function) introduced in [20, 25]asasubsti- tute of a fundamental solution, it is possible however to reduce such a BVP to a system of boundary-domain integral equations, BDIEs, (see e.g. [38, Sect. 18], [43, 44], where the Dirichlet, Neumann, and Robin problems for some PDEs were reduced to indirect BDIEs). However, many questions about their equivalence to the original BVP, solvability, solution uniqueness, and invertibility of corresponding integral operators remained open for rather long time. In [3, 5, 6, 8, 30], the 3D mixed (Dirichlet–Neumann) boundary value problem (BVP) for the stationary diffusion PDE with infinitely smooth variable coefficient on a domain with an © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, pro- vided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Mikhailov Boundary Value Problems (2018) 2018:87 Page 2 of 52 infinitely smooth boundary and a square-integrable right-hand side was reduced to either segregated or united direct boundary-domain integral or integro-differential equations, some of which coincide with those formulated in [29]. Such BVPs appear, for example, in electrostatics, stationary heat transfer, and other diffusion problems for inhomogeneous media. s 1 3 For a function from the Sobolev space H (), < s < , a classical co-normal derivative 2 2 in the sense of traces may not exist. However, the generalised co-normal derivative can be defined in the weak sense, associated with the first Green identity and with an extension s–2 of the corresponding second-order PDE right-hand side to H ()(see [27, Lemma 4.3], [31, Definition 3.1]). Since the extension is non-unique, the co-normal derivative operator appears to be also non-unique and non-linear in u unless a linear relation between u and the PDE right-hand side extension is enforced. This creates some difficulties in formulat- ing the boundary-domain integral equations. These difficulties are addressed in this paper presenting formulation and analysis of di- rect segregated BDIE systems equivalent to the Dirichlet and Neumann boundary value problems, on Lipschitz domains, for the divergent-type PDE with a non-smooth Hölder– s–2 Lipschitz variable scalar coefficient and a general right-hand side from H (), extended s–2 when necessary to H (). This needed a non-trivial generalisation of the third Green identity and its co-normal derivative for such functions, which essentially extends the ap- proach implemented in [3, 5, 6, 8, 30] for the right-hand side from L (), with smooth coefficient and smooth domain boundary. Equivalence of the BDIEs to the original BVP, BDIE solvability, solution uniqueness/non-uniqueness, and the Fredholm properties and invertibility of the BDIE operators are analysed in the Sobolev (Bessel potential) spaces. It is shown that the BDIE operators for the Neumann BVP are not invertible, and appropriate finite-dimensional perturbations are constructed leading to invertibility of the perturbed (stabilised) operators. Some preliminary results in this direction for the infinitely smooth coefficient and domains were presented in [33]. Note that our analysis is mainly aimed not at the boundary-value problems, the proper- ties of which are well known nowadays, but rather at the BDIE systems per se. The analysis is interesting not only in its own rights but is also to be used further on for analysis of con- vergence and stability of BDIE-based numerical methods for PDEs; see, for example, [16, 29, 34, 35, 46–48, 52, 53]. 2 Spaces, co-normal derivatives and boundary value problems n n Let  =  be a bounded open n-dimensional region of R , n ≥ 3, and let  = R \ + – + denote the corresponding exterior domain. For simplicity, we assume that their common boundary ∂ is a simply connected closed Lipschitz surface. Let  denote  ,  or R . 0 + – ∞ n In what follows, D( ):= C ( )and D( ):= {r g : g ∈ D(R )}.Hereand fur- 0 0 0 comp 0 ther on, r denotes the restriction operator on  ; we will also use the equivalent s s s s notation g| := r g.Further, H ( )= H ( )and H (∂)= H (∂)are the Bessel 0 0 0 0 2 2 potential spaces, where s is a real number (see, e.g., [18, 26, 27]). We recall that H s s coincide with the Sobolev–Slobodetski spaces W for non-negative s.By H ( )we s n s n denote the closure of D( )in H (R ). It is a subspace of H (R ), and for Lipschitz s s n s s domains, H ( )= {g : g ∈ H (R ), supp g ⊂  }.By H ( )and H ( )we denote 0 0 0 0 s n s the spaces of restrictions on  of distributions from H (R )and H ( ), respec- 0 0 Mikhailov Boundary Value Problems (2018) 2018:87 Page 3 of 52 tively: s s n H ( ):= r g : g ∈ H R , s s s s H ( ):= r H ( ):= r g : g ∈ H ( ) ⊂ H ( ), 0  0  0 0 • 0 0 endowed by the corresponding infimum norms and the Hilbert structure defined with the s s help of orthogonal projections; see [27,p.77] for H ( ). Note that the space H ( ) 0 0 coincides with the one denoted as L ( )in [41, Eq. (5.2)] and [40, Eq. (2.212)] for s,z 0 p =2. s s n s n Let us introduce the subspace H := {g : g ∈ H (R ), supp g ⊂ ∂ } of H (R )(and of s s s H ( )). By H ( ) wedenotethe closureof D( )in H ( ). 0 0 0 0 Definition 2.1 Let E denote the operator of extension of functions g ∈ H ( ), s ≥ 0, to the whole R by zero outside  .By, e.g.,[27, Lemma 3.32 and Theorem 3.33] (see also s s 1 [31, Theorem 2.7]) the operator E : H ( ) → H ( ) is continuous if 0 ≤ s < ,and 0 0 0 2 1 1 1 we extend it also to the range – < s < defining it for – < s < 0 as (cf. the proof of [31, 2 2 2 Theorem 2.16]) s –s ˚ ˚ E g, v := g, E v , ∀ g ∈ H (), ∀ v ∈ H (). (2.1) 0 0 0 0 Remark 2.2 Note the following known or easily deduced results: s s s 1. There hold the continuous embeddings H ( ) → H ( ) → H ( );see [42, 0 0 0 Eq. (2.123)]. s s 1 2. H ( )= H ( ) for any s >1/2 such that s – is non-integer; see, e.g., [27, 0 0 Theorem 3.3]. s s 3. H ( )= H ( ) for any s ≤ 1/2;see [31, Theorem 2.12]. 0 0 s s s 1 4. H ( )= H ( )= H ( ) for any s <1/2 such that s – is non-integer; see, e.g., 0 0 0 [31, Lemma 2.15]. s s 5. For any s ∈ R, there evidently exists an extension from H ( ) to H ( ),and for 0 0 any s ≥ –1/2,thisextension is unique;see,e.g., [27, Lemma 3.39], [31, Theorem 2.10(i)]. 6. By [31, Theorem 2.16], for any s ∈ (–1/2, 1/2),the extensionfrom s s s s ˚  ˚ H ( )= H ( )= H ( ) to H ( ) is unique and is given by the operator E . 0 0 0 0 • 0 Remark 2.3 Due to Remark 2.2(5), for s ≥ –1/2, the space H ( ) is isometrically iso- morphic to the space H ( ), and sometimes these spaces are identified. Particularly, s s if g , g ∈ H ( ), then denoting by g ˜ , g ˜ ∈ H ( ) the unique distributions such that 1 2 0 1 2 0 g = r g ˜ in  ,wehave g s = g ˜ s and (g , g ) s =(g ˜ , g ˜ ) s .Moreover,if i  i 0 i H ( ) i H ( ) 1 2 H ( ) 1 2 H ( ) • 0 0 • 0 0 ˚ ˚ s ∈ (–1/2, 1/2), then by Remark 2.2(6), g ˜ = E g hence implying g s = E g . i  i i  i 0 H ( ) 0 H ( ) • 0 0 There is no such isomorphism for s < –1/2 since in such a case the extension from s s H ( )to H ( ) is not unique. However, due to the definition of the spaces, there is still 0 0 s s s an isometric isomorphism between the space H ( ) and the quotient space H ( )/H . 0 0 • ∂ Definition of the space H ( ), Remark 2.2,and Remark 2.3 imply the following asser- tion. Mikhailov Boundary Value Problems (2018) 2018:87 Page 4 of 52 Corollary 2.4 The following restriction operators are isomorphisms: s s r : H ( ) → H ( ), – ≤ s, (2.2) 0 0 0 • 1 1 s s s r : H ( ) → H ( )= H ( ), – < s < , (2.3) 0 0 0 0 • 2 2 s s s r : H ( )/H → H ( ), s <– . (2.4) 0 0 0 ∂ • –1 The inverse to the operator (2.3) is r = E ; see Definition 2.1. m,θ Definition 2.5 For a non-negative integer m and 0 < θ ≤ 1, let C ( )denote the Hölder–Lipschitz space in the closed domain  . Similar to [32, Definition 3.1], g ∈ ( )for μ ≥ 0means that + 0 g ∈ L ( ) when μ =0; ∞ 0 μ–1,1 g ∈ C ( ) when μ is a positive integer; m,θ+ g ∈ C ( ) for some  >0 when μ = m + θ,where m is a non-negative integer, and 0< θ <1. Employing this definition, Theorem 7.2 from Sect. 7 can be reformulated as follows. |σ | n σ Theorem 2.6 Let  be an open set in R , σ ∈ R, v ∈ H ( ), and g ∈ C ( ). Then g 0 0 + 0 σ σ σ is a multiplier in H ( ), i.e., gv ∈ H ( ) for every v ∈ H ( ), and the corresponding 0 0 0 norm estimate holds. Let us denote ∂ := ∂ := ∂/∂x (j = 1,2,..., n), ∇ =(∂ , ∂ ,..., ∂ ). Let j x j 1 2 n 0< a ≤ a(x) ≤ a < ∞ for almost every x ∈  . (2.5) min max ± We consider the scalar elliptic differential equation, which can be written in the following strong form if u and a are sufficiently smooth: Au(x):= A(x, ∇)u(x):= ∇· a(x)∇u(x) = f (x), x ∈  , (2.6) where u is an unknown function and f is a given function in  . |s–1| For u ∈ H ( ), 1/2 < s <3/2, and a ∈ C ( ), the partial differential operator A is ± + ± understood in the sense of distributions: Au, v := –E (u, v), ∀v ∈ D( ), (2.7) ± ± where E (u, v):= a∇u, ∇v := a∂ u, ∂ v , i i ± ± ± i=1 and the duality brackets g, · denote value of a linear functional (distribution) g extend- ing the usual L dual product. If s =1, then E (u, v)= a(x)∇u(x) ·∇v(x) dx. ± Mikhailov Boundary Value Problems (2018) 2018:87 Page 5 of 52 2–s Since the set D( )is dense in H ( ), (2.7) defines, due to Theorem 2.6 (see, e.g., ± ± s s–2 2–s ∗ [32, Theorem 3.4]), the continuous linear operator A : H ( ) → H ( )=[H ( )] , ± ± ± where s 2–s Au, v := –E (u, v), ∀ u ∈ H ( ), v ∈ H ( ). (2.8) ± ± ± ± s s–2 2–s ∗ Let us also consider the operator A : H ( ) → H ( )=[H ( )] (see [31, ± ± ± Eq. (3.5)], [32, Eq. (5.1)]) defined by ˇ ˇ A u, v := –E (u, v):=– E (a∇u), ∇v ± ± ± ± ˚ ˚ =– E (a∇u), ∇v = ∇· E (a∇u), v e n  e n ± ± R R s 2–s = ∇· E (a∇u), v , ∀ u ∈ H ( ), v ∈ H ( ), (2.9) ± ± 2–s n which is evidently continuous. Here v ∈ H (R )is such that r v = v. Evidently, weak e  e definition (2.9) can be also written (in the strong-looking form) as A u = ∇· E r [a∇u]. (2.10) ± ± ± s s–2 For any u ∈ H ( ), the functional A u belongs to H ( ) and is a specific extension ±  ± s–2 s–2 of the functional Au ∈ H ( ); recall that the functional Au ∈ H ( )is defined on ± ± 2–s 2–s H ( ), whereas the functional A u is defined on H ( ). ±  ± Remark 2.7 Note also that Definition 2.1 for E and definition (2.9)imply that ˇ ˇ ˇ ˇ A u, v =–E (u, v)=–E (v, u)= u, A v , ± ± ± ± ± ± s 2–s ∀ u ∈ H ( ), v ∈ H ( ), 1/2 < s < 3/2. ± ± From the trace theorem (see, e.g., [11, 12, 26, 27]) for u ∈ H ( ), 1/2 < s < 3/2, it follows s– ± ± ± + – that γ u ∈ H (∂), where γ = γ is the trace operator on ∂ from  .If γ u = γ u, –1 –1 s– s n then we will sometimes write just γ u.Let also γ := γ : H (∂) → H (R )denote –1 a (non-unique) continuous right inverse to the trace operator γ , i.e., γγ w = w for any 1 1 s– ± –1 s– 2 2 w ∈ H (∂). Hence also γ γ w = w for any w ∈ H (∂). s c± For u ∈ H ( ), s > ,and a ∈ C( ), we denote by T the corresponding classical ± ± (strong) co-normal derivative operators on ∂ in the sense of traces: c± ± T u(x):= a(x)ν(x) · γ ∇u(x)= a(x)∂ u(x), x ∈ ∂, (2.11) where ν(x)= ν (x) is the outward to  unit normal vector at the point x ∈ ∂,and we will c c+ c– sometimes write T u(x)if T u(x)= T u(x). However, the classical co-normal derivative is, generally, not well defined if u ∈ H ( ), 1/2 < s < 3/2,(seeanexamplein[33,Ap- pendix A] of a function from H (), wherethe classicalnormalderivativedoesnot exist at boundary points). Inspired by the first Green identity for smooth functions, we can define the generalised co-normal derivative (cf., e.g., [27, Lemma 4.3]), [31, Definition 3.1], [32, Definition 5.2]). Mikhailov Boundary Value Problems (2018) 2018:87 Page 6 of 52 |s–1| Definition 2.8 Let 1/2 < s < 3/2, u ∈ H ( ), a ∈ C ( ), and r Au = r f for some ± + ±   ± ± ± s–2 ± s– ˜ ˜ f ∈ H ( ). Then the generalised co-normal derivatives T (f ; u) ∈ H (∂)are de- ± ± ± fined in the weak form as ± –1 –1 ˜ ˜ ˇ ± T (f ; u), w := f , γ w + E u, γ w ± ± –1 –s ˜ ˇ = f – A u, γ w , ∀ w ∈ H (∂), (2.12) i.e., ± –1 ˜ ˜ ˇ T (f , u):= ± γ (f – A u). (2.13) ± ± If a ≡ 1, then A = ,and T (f ; u)become generalised normal derivatives denoted as T (f ; u). 1 1 –1 ∗ –t n –t+ –1 t– t n 2 2 The operator (γ ) : H (R ) → H (∂)is dual to γ : H (∂) → H (R )and is –1 ∗ –1 t– –t n defined as (γ ) ψ, w := ψ, γ w n for any w ∈ H , ψ ∈ H (R ), 1/2 < t < 3/2. In ∂ R (2.13)itwas employed for t =2 – s. Theorem 2.9 (Lemma 4.3 in [27], Theorem 3.2 in [31], and Theorem 5.3 in [32]) Un- der the hypotheses of Definition 2.8, the generalised co-normal derivatives T u(f ; u) are –1 independent of (non-unique) choice of the operator γ , and we have the estimate ˜ ˜ T (f ; u) 3 ≤ C u s + C f s–2 (2.14) ± 1 H ( ) 2 ± H ( ) s– ± ± H (∂) and the first Green identity in the form ± ± ˜ ˜ ˇ ± T (f ; u), γ v = f , v + E (u, v) ± ± ± ± 2–s ˜ ˇ = f – A u, v , ∀ v ∈ H ( ). (2.15) ±   ± ± ± As follows from Definition 2.8, the generalized co-normal derivative is nonlinear with re- ˜ ˜ spect to u for fixed f but still linear with respect to the couple (f , u), i.e., for any complex ± ± numbers α and α , 1 2 ± + ± ± ˜ ˜ ˜ ˜ α T (f ; u )+ α T (f ; u )= T (α f ; α u )+ T (α f ; α u ) 1 1± 1 2 2± 2 1 1± 1 1 2 2± 2 2 ˜ ˜ = T (α f + α f ; α u + α u ). 1 1± 2 2± 1 1 2 2 Let us also define some subspaces of H ( ); see [11, 15, 31, 32]. s ∗ Definition 2.10 Let s ∈ R,and let A : H ( ) → D ( ) be a linear operator. For t ∈ R, ∗ ± ± we introduce the space s,t s t H ( ; A ):= g : g ∈ H ( ), A g ∈ H ( ) ± ∗ ± ∗ ± 2 2 1/2 s,t endowed with the norm g := ( g + A g ) and the correspond- H ( ;A ) s ∗ t ± ∗ H ( ) ± H ( ) • ± ing inner product. Mikhailov Boundary Value Problems (2018) 2018:87 Page 7 of 52 s,t Definition 2.11 Let  be either  or  .ByRemark 2.3,if g ∈ H ( ; A )for some 0 + – 0 ∗ 1 t ˜  ˜ s ∈ R and t ≥ – , then there exists a unique distribution f ∈ H ( )such that r f = A g, 0  ∗ 2 0 –1 s,t t ˜ ˜ ˜ ˜ and hence f = A g,where A := r A . The operator A : H ( ; A ) → H ( ), ∗ ∗ ∗ ∗ 0 ∗ 0 0 0  0 which is continuous by Corollary 2.4, is called the canonical extension of the operator s,t t 1 1 ˜ ˚ A : H ( ; A ) → H ( ), and moreover, if – < t < ,then A = E A . ∗ 0 ∗ 0 ∗  ∗ • 2 2 0 0 We will mostly use the operators A or as A in the definition. Note that since Au = 1 1 s,– s,– 2 2 2 u + ∇a ·∇u,for 1/2 < s < 3/2, we have H ( ; A)= H ( ; )if a ∈ C ( ), with 0 0 + 0 equivalent norms. Let us now define the canonical conormal derivative; see [32, Definition 6.5]. |s–1| s,– Definition 2.12 For u ∈ H ( ; A)and a ∈ C ( ), 1/2 < s < 3/2, we define the ± + ± s– canonical co-normal derivatives T u ∈ H (∂)as ± –1 –1 –1 ˜ ˜ ˇ ± T u, w := A u, γ w + E u, γ w = A u – A u, γ w ± ± ± ± ± ± –1 –s ˜ ˇ = γ (A u – A u), w ∀ w ∈ H (∂), (2.16) ± ± i.e, ± –1 ˜ ˇ T u := ± γ (A u – A u). (2.17) ± ± ± ± If a ≡ 1, then T u becomes the canonical normal derivative denoted as T u. Theorem 2.13 (Theorem 3.9 in [31]and Theorem6.6 in [32]) Under the hypotheses of Definition 2.12, the canonical co-normal derivatives T uare independent of (non-unique) 1 3 –1 ± s,– s– 2 2 choice of the operator γ , the operators T : H ( ; A) → H (∂) are continuous, and the first Green identity holds in the form ± ± ˜ ˇ ± T u, γ v = A u, v + E (u, v) ± ± ± 2–s ˜ ˇ = A u – A u, v , ∀ v ∈ H ( ). (2.18) ± ± ± The canonical co-normal derivatives in Definition 2.12 are completely defined by the func- tion u and operator A only and do not depend explicitly on the right-hand sides f ,un- like the generalised co-normal derivatives defined in (2.15), whereas the operators T are linear in u. Note that the canonical co-normal derivatives coincide with the classical co- ± c± normal derivatives T u = T u if the latter do exist (see [32, Corollaries 6.11 and 6.14]), which is generally not the case for the generalised conormal derivatives even for smooth ˜ ˜ functions u,unless f = A u is chosen. Thus the canonical conormal derivative is a con- tinuous extension of the classical conormal derivative. |s–1| s,– Let 1/2 < s < 3/2 and a ∈ C ( ). If u ∈ H ( ; A), then Definitions 2.8 and 2.12 + ± ± s–2 imply that the generalised co-normal derivative for arbitrary extensions f ∈ H ( )of ± ± the distributions r Au can be expressed as ± ± –1 ˜ ˜ ˜ T (f ; u)= T u ± γ (f – A u). (2.19) ± ± ± Mikhailov Boundary Value Problems (2018) 2018:87 Page 8 of 52 s 2–s,– If u ∈ H ( )and v ∈ H ( ; A), then swapping over the roles of u and v in (2.18), ± ± we obtain the first Green identity for v: ± ± ˇ ˜ ± T v, γ u = E (v, u)+ A v, u . (2.20) ± ± ± s–2 ˜ ˜ If, in addition, r Au = r f ,where f ∈ H ( ), then subtracting (2.20)from(2.15) ± ± ± ± ± ˇ ˇ and taking into account that E (u, v)= E (v, u)by Remark 2.7, we obtain the following ± ± second Green identity: ± ± ± ± ˜ ˜ ˜ f , v – A v, u = ± T (f ; u), γ v ∓ T v, γ u . ±    ± ± ± ± ∂ ∂ 1 1 s,– 2–s,– 2 2 If, finally, u ∈ H ( ; A)and v ∈ H ( ; A), then we arrive at the familiar form of ± ± the second Green identity for the canonical extension A of the operator A and the canon- ical co-normal derivatives ± ± ± ± ˜ ˜ A u, v – A v, u = ± T u, γ v ∓ T v, γ u . (2.21) ± ± ± ± ∂ ∂ 3 Parametrix and potential type operators on Lipschitz domains Recall that unless stated otherwise, we will assume that  =  . We will say that a function P(x, y)of two variables x, y ∈ R is a parametrix (the Levi function) for the operator A(x, ∂ )in R if (see, e.g., [19, 20, 25, 29, 38, 43, 44]) A(x, ∇ )P(x, y)= δ(x – y)+ R(x, y), (3.1) where δ(·) is the Dirac distribution, and R(x, y) possesses a weak (integrable) singularity at x = y, i.e., –κ R(x, y)= O |x – y| with κ < n. (3.2) n/2 2π n Let ω = denote the area of the unit sphere in R . It is well known that the function (n/2) –1 P (x, y)= , x, y ∈ R , (3.3) n–2 (n –2)ω |x – y| is the fundamental solution of the Laplace equation, i.e., P (x, y)= P (x, y)= δ(x – y). It is easy to see that for the operator A(x, ∂ ) given by the left-hand side in (2.6), the function 1 –1 P(x, y)= P (x, y)= , x, y ∈ R , (3.4) n–2 a(y) (n –2)ω a(y)|x – y| is a parametrix, whereas the corresponding remainder function is R(x, y)= ∇a(x) ·∇ P(x, y)= ∇a(x) ·∇ P (x, y) x x a(y) (x – y) ·∇a(x) = , x, y ∈ R , (3.5) ω a(y)|x – y| n Mikhailov Boundary Value Problems (2018) 2018:87 Page 9 of 52 1 n and if a ∈ C (R ), then it satisfies estimate (3.2)a.e.with κ = n –1. Note also that A(y, ∇ )P(x, y)= δ(x – y)+ R (x, y), (3.6) y ∗ where R (x, y)=–∇ · P(x, y)∇a(y) ∗ y (ln a(y)) (x – y) ·∇a(y) = – , x, y ∈ R . (3.7) n–2 n (n –2)ω |x – y| ω a(y)|x – y| n n Evidently, the parametrix P(x, y)given by (3.4) is related to the fundamental solution to the operator A(y, ∇ ):= a(y) with “frozen” coefficient a(y), and A(y, ∇ )P(x, y)= δ(x – y). x x x Note that parametrix (3.4) and remainders (3.5)and (3.7) are not smooth enough for the corresponding potential operators to be directly treated as in [27], which thus need some additional consideration. For g ∈ D(R ) and sufficiently smooth coefficient a, the parametrix-based volume po- tential operator and the remainder potential operator corresponding to parametrix (3.4) and remainders (3.5)and (3.7)for y ∈ R are Pg(y):= P(·, y), g = P(x, y)g(x) dx, (3.8) Rg(y):= R(·, y), g = R(x, y)g(x) dx, (3.9) R g(y):= R (·, y), g = R (x, y)g(x) dx, (3.10) ∗ ∗ ∗ and from (3.1)–(3.10)weobtain, PAg = g + Rg, APg = g + R g in R . (3.11) For the function g defined on a domain  ⊂ R , e.g., g ∈ D( ), the corresponding + + potentials for y ∈  are Pg(y):= P(·, y), g = P(x, y)g(x) dx, (3.12) Rg(y):= R(·, y), g = R(x, y)g(x) dx, (3.13) R g(y):= R (·, y), g = R (x, y)g(x) dx. (3.14) ∗ ∗ ∗ From definitions (3.4), (3.5), and (3.7) we can obtain representations of the parametrix- based potential operators in terms of their counterparts for a = 1 (i.e., associated with the Laplace operator ; see, e.g., [21]), which we equip with the subscript (see [3]): 1 1 ∇a Pg = P g, Rg =– ∇· P (g∇a), R g =–∇· P g , (3.15) a a a 1 1 ∇a Pg = P g, Rg =– ∇· P (g∇a), R g =–∇· P g . (3.16) a a a Mikhailov Boundary Value Problems (2018) 2018:87 Page 10 of 52 Hence (aPg)= g in R , (aPg)= g in . (3.17) Employing relations (3.16) and the well-known properties of the operator P as the pseudo-differential operator of order –2 together with Theorem 2.6, definitions (3.8)– s n s (3.10) can be extended to g ∈ H (R ), g ∈ H () and lower-smoothness coefficient a.For s s g ∈ H ()and g ∈ H (), the potentials P, R, R defined on functions (or distributions) having support on  are understood as Pg := r Pg, Rg := r Rg, R g := r R g, g ∈ H (), s ∈ R; (3.18) ∗  ∗ ˚ ˚ ˚ Pg := r PE g, Rg := r RE g, R g := r R E g, g ∈ H (), s >– . (3.19) ∗  ∗ To prove mapping properties of the parametrix-based volume potential operators in Sobolev spaces, we first provide some well-known results for the classical Newtonian vol- ume potential associated with the Laplace operator. Lemma 3.1 Let  be a bounded Lipschitz domain in R . The following operators are con- tinuous: s–2 n s n n μP : H R → H R , s ∈ R, ∀ μ ∈ D R ; (3.20) s–2 s P : H () → H (), s ∈ R; (3.21) 3 5 s–2 s P : H () → H (), < s < ; (3.22) 2 2 1 3 s–2 s,– P : H () → H (; ), s ≥ ; (3.23) 1 1 3 + s–2 s– γ P : H () → H (∂), < s < ; (3.24) 2 2 + s 1 γ P : H () → H (∂), – < s; (3.25) + s T P : H () → L (∂), – < s; (3.26) + s T P : H () → L (∂), – < s. (3.27) 1 3 s–2 s–2 ˜  ˜  ˜ ˜ If < s < , f ∈ H (), and f ∈ H () is such that r f = r f , then there exist constants 0  0 2 2 C , C >0 such that 0 1 ˜ ˜ ˜ ˜ T (f ; P f ) 3 ≤ C f s–2 + C f s–2 . (3.28) 1  2 0 s– H () H () H (∂) Proof Operator (3.20) and hence (3.21)are continuous since P is a pseudo-differential operator of order –2. The continuity of operator (3.22) follows from the first relation in (3.19)for P and P and from (3.21). Since P g = g in , the continuity of operator (3.21) implies that of operator (3.23). Mikhailov Boundary Value Problems (2018) 2018:87 Page 11 of 52 The continuity of operator (3.24) is implied by that of operator (3.21) and the trace the- orem for Lipschitz domains; see, e.g., [11, Lemma 3.6] and [27, Theorem 3.38]. The con- tinuity of operator (3.25) follows from that of operator (3.22) and from, e.g., [54], [31, 1 1 1 Lemma 2.5] for – < s < , and then by the embedding argument for s ≥ . 2 2 2 The continuity of operators (3.26)and (3.27)isimplied by that of (3.21)and (3.22), re- spectively, and by [31, Corollary 3.14] since s +2 > in both cases. Estimate (3.28) follows ˜ ˜ from the continuity of operator (3.21), relation P f = f , and estimate (2.14). Now the following mapping properties of the parametrix-based operators can be ob- tained. Theorem 3.2 Let  be a bounded Lipschitz domain in R . The following operators are continuous: s–2 n s n |s| n n μP : H R → H R , s ∈ R, a ∈ C R , ∀ μ ∈ D R ; (3.29) s–2 s |s| P : H () → H (), s ∈ R, a ∈ C (); (3.30) 3 5 s–2 s s P : H () → H (), < s < , a ∈ C (); (3.31) 2 2 1 3 s–2 s,– s P : H () → H (; A), ≤ s, a ∈ C (); (3.32) s–1 n s n |s–1|+1 n n μR : H R → H R , s ∈ R, a ∈ C R , ∀ μ ∈ D R ; (3.33) 1 3 s–1 s |s–1|+1 R : H () → H (), < s < , a ∈ C (); (3.34) 2 2 1 3 s s s R : H () → H (), < s < , a ∈ C (); (3.35) 2 2 1 1 3 s s,– R : H () → H (; A), < s < , a ∈ C (); (3.36) 2 2 s n s+1 n |s+2|+1 n n μR : H R → H R , s ∈ R, a ∈ C R , ∀ μ ∈ D R ; (3.37) s s+1 |s+2|+1 R : H () → H (), s ∈ R, a ∈ C (); (3.38) 3 1 s σ 2 R : H () → H (), – < s, a ∈ C (), for some σ >– ; (3.39) ∗ + 2 2 1 1 3 + s–2 s– s γ P : H () → H (∂), < s < , a ∈ C (); (3.40) 2 2 + s 1 2 γ P : H () → H (∂), – < s, a ∈ C (); (3.41) 1 1 3 + s s– s γ R : H () → H (∂), < s < , a ∈ C (); (3.42) 2 2 + s 2 T P : H () → L (∂), – < s, a ∈ C (); (3.43) 2 + + s 2 T P : H () → L (∂), – < s, a ∈ C (); (3.44) 2 + 3 1 3 + s s– 2 T R : H () → H (∂), < s < , a ∈ C (). (3.45) 2 2 Moreover, operators (3.35), (3.36), (3.42), and (3.45) are compact. Mikhailov Boundary Value Problems (2018) 2018:87 Page 12 of 52 1 3 s s–2 s–2 ˜ ˜ ˜ ˜ If < s < , a ∈ C (), f ∈ H (), and f ∈ H () is such that r f = r APf , then 0  0 2 2 there exist constants C , C >0 such that 0 1 ˜ ˜ ˜ ˜ T (f ; Pf ) 3 ≤ C f s–2 + C f s–2 . (3.46) 0 1  2 0 s– H () H () H (∂) Proof The continuity of operators (3.29)–(3.31) is implied by the first relations in (3.15) and (3.16) and by the continuity of operators (3.20)–(3.22) together with Theorem 2.6. The continuity of operators (3.30)and (3.31)and Remark 2.2(4) imply that of operator 1 3 3 3 2 2 (3.32)for s > .Let us nowprove (3.32)for s = .For g ∈ H (), we have, Pg ∈ H () 2 2 due to (3.30), whereas APg = ∇· a∇ P g = g – ∇· (∇ ln a)P g in , (3.47) wherewehavetaken into account that P g = g. The first term in the right-hand side of 2 σ σ (3.47)belongs to H (), whereas the second term belongs to H ()= H ()for some σ ∈ (–1/2, 1/2) (cf. item 4 in Remark 2.2)since ∇a ∈ C ()and a ≥ a >0, which com- + min pletes the proof of the continuity of operator (3.32). The continuity of operator (3.33) follows from the second relation in (3.15)togetherwith Theorem 2.6 and the continuity of operator (3.20). Indeed, let us take arbitrary μ ∈ D(R ), let B be a ball such that supp μ ⊂ B ,and let μ ∈ D(R )be such that μ =1 in B .Then μ μ 1 1 μ s–1 n for any g ∈ H (R ), we have μRg s n = ∇· μ P (g∇a) ≤ c ∇· μ P (g∇a) H (R ) 1 1 1 s n H (R ) s n H (R ) ≤ c μ P (g∇a) ≤ c g∇a s–1 n ≤ c g s–1 n , (3.48) 2 1 3 4 s+1 n H (R ) H (R ) H (R ) where c are positive constants (depending on μ, μ ,and a), and we took into account that i 1 |s–1|+1 |s| n n C (R ) ⊂ C (R )since |s|≤|s –1| +1. + + To prove the continuity of operator (3.34), we similarly employ the second relation in (3.16) together with Theorem 2.6 and the continuity of operator (3.22). Then we obtain s–1 for any g ∈ H (), 1/2 < s < 3/2, and some positive constants c : Rg = ∇· P (g∇a) ≤ c ∇· P (g∇a) H () H () H () ≤ c P (g∇a) ≤ c g∇a s–1 ≤ c g s–1 . (3.49) 3 4 s+1 H () H () H () Let us prove the continuity and compactness of operator (3.35). For 1 ≤ s < ,wehave s = |s –1| + 1, and then the continuity of operator (3.34) implies the continuity and com- pactness of (3.35). For < s < 1, we need a sharper estimate of the norm g∇a s–1 . H () First, by Definition 2.5 the inclusion a ∈ C () implies that there exists t ∈ (s,1) such 0,t t t that a ∈ C ()= B ()= F (); see, e.g., Proposition in [45, Sect. 2.1.2], and hence ∞,∞ ∞,∞ t–1 ∇a ∈ F (). Then, by Theorems 1 from [45, Sect. 4.4.3] we have ∞,∞ g∇a ≤ C ∇a g σ t–1 t–1 H () F () F () ∞,∞ 2,∞ ≤ C a g σ , ∀ σ ∈ (1 – t, s). (3.50) 0,t H () C () Mikhailov Boundary Value Problems (2018) 2018:87 Page 13 of 52 On the other hand, by (3.49), item (ii) of Proposition from [45, Sect. 2.2.1], and (3.50)we obtain Rg s ≤ c g∇a = c g∇a s–1 s–1 H () 3 3 H () F () 2,2 ≤ C g∇a t–1 ≤ C C a g σ . 0,t 1 1 H () F () C () 2,∞ σ s Thus the operator R : H () → H () is continuous, which implies the continuity and, by the Rellich compact embedding theorem, also the compactness of operator (3.35)for < s <1. Let us prove the continuity of operator (3.36). Since a ∈ C (), by Definition 2.5 there 1, + 1 1 exists  >0 such that a ∈ C (), and let us choose any σ ∈ ( , min{s, + }). By the 2 2 σ s continuity of (3.34) the operator R : H () → H () is continuous. Now let us prove σ 2 that the operator AR : H () → H () is continuous as well. Indeed, for some positive constants c ,wehave ARg 1 ≤ ARg σ –1 – H () H () ≤ c ARg σ –1 0 H () = c ∇· a∇ ∇· P (g∇a) σ –1 H () = c –∇· (∇ ln a)∇· P (g∇a) + ∇· P (g∇a) σ –1 H () ≤ c –(∇ ln a)∇· P (g∇a)+(g∇a) H () ≤ c a 1 P (g∇a) + c g∇a 1 H () σ +1 1, + H () σ σ ≤ c g∇a σ –1 + c g∇a ≤ c g∇a 3 1 H () 4 H () H () ≤ c a g σ . 5 H () 1, + C 2 σ s,– Hence we proved the continuity of the operator H () → H (; A), which implies that of operator (3.36) and by the Rellich compact embedding theorem also its compactness. The continuity of operator (3.37)isimplied by thelastrelationin(3.15), the continuity of operator (3.20), and Theorem 2.6 in the chain of inequalities analogous to (3.48). Similarly, the continuity of operator (3.38)isimplied by thelastrelationin(3.16), the continuity of operator (3.21), and Theorem 2.6. The continuity of operator (3.39)isimplied by that of 2 1,1/2+ (3.38)since a ∈ C () implies that there exists  >0 such that a ∈ C (), and we can 3 3 take σ ∈ ( , min{s +1, + }). 2 2 The continuity of operator (3.40) is implied by that of operator (3.30) and the trace theo- rem for Lipschitz domains; see, e.g., [11, Lemma 3.6], [27, Theorem 3.38]. The continuity 1 1 of operator (3.41)for – < s <– +  with any sufficiently small  > 0 follows from that 2 2 of operator (3.31) together with, e.g., [54], [31, Lemma 2.5] and then by the embedding argument for all s >– . Similarly, the continuity of operators (3.43)and (3.44)isimplied by that of (3.30)and (3.31), respectively, and by [31, Corollary 3.14] since s +2 > in the both cases. Mikhailov Boundary Value Problems (2018) 2018:87 Page 14 of 52 The continuity and compactness of operators (3.42)and (3.45) are implied by those of operators (3.35)and (3.36), the trace theorem for Lipschitz domains, and Theorem 2.9. Estimate (3.46) follows from the continuity of operator (3.30) and estimate (2.14). The parametrix-based single- and double-layer surface potential operators are defined as Vg(y):=– P(x, y)ψ(x) dS , y ∈/ ∂, (3.51) Wg(y):=– T x, n(x), ∂ P(x, y) ϕ(x) dS , y ∈/ ∂, (3.52) x x where the integrals are understood as duality forms if ψ and ϕ are not integrable. Partic- 1 1 1 3 –s –s 2 2 ularly, for ψ ∈ H (∂)and ϕ ∈ H (∂), < s < ,wehave 2 2 V ψ(y):= – γ P(·, y), ψ =– P(·, y), γ ψ ∂ R ∗ ∗ =– Pγ ψ(y)=– P γ ψ(y), (3.53) a(y) c c∗ W ϕ(y):= – T P(·, y), ϕ =– P(·, y), T ϕ ∂ R c∗ c∗ =– PT ϕ(y)=– P T ϕ(y), (3.54) a(y) ∗ c∗ –1 where γ ψ and T ϕ are well defined for any ψ ∈ H (∂), ϕ ∈ L (∂), and a ∈ L (∂), 2 ∞ in the sense of distributions, as ∗ n γ ψ, φ := ψ, γφ , ∀ φ ∈ D R ,and n ∂ c∗ c c n T ϕ, φ := ϕ, T φ = ϕ, aT φ , ∀ φ ∈ D R , R ∂ ∂ ∗ c∗ which evidently implies that supp γ ψ ⊂ ∂ and supp T ϕ ⊂ ∂.Moreover, 1 1 1 3 ∗ –s –s c∗ –s –s–1 2 2 γ : H (∂) → H , T : H (∂) → H , < s < , (3.55) ∂ ∂ 2 2 are the continuous operators adjoint, respectively, to the continuous trace operator γ : s n s– c H (R ) → H (∂) and to the continuous classical conormal derivative operator T : loc s+1 n s– c c∗ H (R ) → H (∂); for the continuity of T and T ,itisalsoassumed that a ∈ loc s– C (∂). When a =1, formulas (3.51)and (3.52) define the corresponding harmonic potentials, which we denote as V and W , respectively. From definitions (3.51)and (3.52), similar to (3.15)–(3.16), we have (cf. [3]) 1 1 Vg = V g, Wg = W (ag). (3.56) a a Hence (aVg)=0, (aWg)=0 in  . (3.57) ± Mikhailov Boundary Value Problems (2018) 2018:87 Page 15 of 52 We will mainly need the restrictions of the layer potentials to , i.e., r V and r W,but will often omit the restriction operator r if this is clear from the context. The mapping properties and jump relations for the single- and double-layer potentials are well known for the case a = const and were extended to the case of infinitely smooth boundary and variable coefficient a(x)in [3, 5]. Before proving the corresponding prop- erties for the parametrix-based potentials on Lipschitz domains, we further collect the following well-known mapping and jump properties for the harmonic potentials on Lips- chitz domains. Theorem 3.3 Let  be a bounded Lipschitz domain in R . 1 3 n (i) If ≤ s ≤ , then the following operators are continuous for any μ ∈ D(R ): 2 2 s– s n μV : H (∂) → H R , (3.58) 1 1 s– s s– s 2 2 r W : H (∂) → H (), μr W : H (∂) → H ( ). (3.59) 1 3 (ii) If < s < , then the following operators are continuous: 2 2 3 1 1 1 ± s– s– ± s– s– 2 2 2 2 γ V : H (∂) → H (∂), γ W : H (∂) → H (∂), (3.60) 3 3 1 3 ± ± s– s– s– s– 2 2 2 2 T V : H (∂) → H (∂), T W : H (∂) → H (∂), (3.61) 1 3 1 3 s– s– 2 2 (iii) If < s < , then, for any ϕ ∈ H (∂) and ψ ∈ H (∂), the following jump 2 2 properties hold: + – + – γ V ψ – γ V ψ =0, γ W ϕ – γ W ϕ =–ϕ, (3.62) + – + – T V ψ – T V ψ = ψ, T W ϕ – T W ϕ = 0. (3.63) Proof Items (i) and (ii) follow, e.g., from [11, Theorem 1(i,ii) and Remark], [22–24, 51](see also [27, Theorem 6.12]) if we take into account that the canonical co-normal derivative operators in (3.61)are well defined since V =0and W =0in  .The jump properties of item (iii) for s = 1 are implied, e.g., by [11, Lemma 4.1]; see also [27, Theorem 6.11]. 3 1 Hence they evidently hold if 1 ≤ s < and by the density argument also if < s <1. 2 2 Theorem 3.3 implies the following assertion. 1 3 Corollary 3.4 Let ∂ be a compact Lipschitz boundary, and let < s < . The following 2 2 operators are continuous: 3 1 + – s– s– 2 2 V := γ V = γ V : H (∂) → H (∂), (3.64) 1 1 1 + – s– s– 2 2 W := γ W + γ W : H (∂) → H (∂), (3.65) 1 3 3 + – s– s– 2 2 W := T V + T V : H (∂) → H (∂), (3.66) 1 3 + – s– s– 2 2 L := T W = T W : H (∂) → H (∂). (3.67) Mikhailov Boundary Value Problems (2018) 2018:87 Page 16 of 52 Employing relations (3.56), Theorem 3.3,and Theorem 2.6, we obtain the following mapping properties for the parametrix-based potentials on Lipschitz domains. Theorem 3.5 Let  be a bounded Lipschitz domain. 1 3 (i) The following operators are continuous if ≤ s ≤ , 2 2 s– s n s n n μV : H (∂) → H R , a ∈ C R , ∀ μ ∈ D R ; (3.68) s– s s r W : H (∂) → H (), a ∈ C (); (3.69) s– s s n μr W : H (∂) → H ( ), a ∈ C ( ), ∀ μ ∈ D R . (3.70) – – – + 1 3 2 (ii) The following operators are continuous if < s ≤ and a ∈ C (): 2 2 3 1 s– s,– 2 2 r V : H (∂) → H (; A); (3.71) s,– s– 2 n μr V : H (∂) → H ( ; A), ∀ μ ∈ D R ; (3.72) loc 1 1 s– s,– 2 2 r W : H (∂) → H (; A); (3.73) 1 1 s– s,– n 2 2 μr W : H (∂) → H ( ; A), ∀ μ ∈ D R . (3.74) 1 3 (iii) The following operators are continuous if < s < : 2 2 3 1 ± s– s– s 2 2 γ V : H (∂) → H (∂), a ∈ C ( ); (3.75) 1 1 ± s– s– s 2 2 γ W : H (∂) → H (∂), a ∈ C ( ); (3.76) 3 3 ± s– s– 2 2 T V : H (∂) → H (∂), a ∈ C ( ); (3.77) 1 3 ± s– s– 2 2 2 T W : H (∂) → H (∂), a ∈ C ( ). (3.78) + ± Proof Relations (3.56), Theorem 3.3(i), and Theorem 2.6 immediately imply the continuity of operators (3.68)and (3.69). Further, if a ∈ C (), then there exists  >0 such that a ∈ 1 3 1, + 1 3 s– 1 1 2 2 C (). For < s ≤ , g ∈ H (), and any σ ∈ ( , min{s, + }), we have 2 2 2 2 AVg σ –1 = ∇· a∇ V g = ∇· (∇ ln a)V g σ –1 H () H () a σ –1 H () ≤ (∇ ln a)V g ≤ C a V g σ H () 1, + C 2 () ≤ C a 1 V g , H () 1, + C () wherewehavetaken into accountthat V g =0 in .Hence,along with continuity σ –1 of operator in (3.58), this implies AVg ∈ H ()and thus, by Remark 2.2(4), r AVg ∈ σ –1 2 H () ⊂ H () with the corresponding norm estimate, from which the continuity of operator (3.71) follows. The continuity of operator (3.73)isprovedinasimilarfashion. The continuity of operators (3.70), (3.72), and (3.74) immediately follows from the con- tinuity of their counterparts for the interior domain. Mikhailov Boundary Value Problems (2018) 2018:87 Page 17 of 52 The continuity of operators (3.75)and (3.76) for the potential traces is implied by the continuity of operators (3.68)–(3.70) and the trace theorem, whereas the continuity of op- erators (3.77)and(3.78) for the potential co-normal derivatives is implied by the continuity of operators (3.71)–(3.74)and Theorem 2.9. Now we can prove the jump properties for the parametrix-based potentials on Lipschitz domains. 1 3 s– Theorem 3.6 Let ∂ be a compact Lipschitz boundary, < s < , ϕ ∈ H (∂), and ψ ∈ 2 2 s– H (∂). Then + – + – s n γ V ψ – γ V ψ =0, γ W ϕ – γ W ϕ =–ϕ, if a ∈ C R ; (3.79) + – + – 2 n T V ψ – T V ψ = ψ, T W ϕ – T W ϕ =(∂ a)ϕ, if a ∈ C R . (3.80) Proof Relations (3.56)and (3.62)along with Theorem 2.6 immediately imply jump rela- tions (3.79). To prove the first jump relation in (3.80), we generalise to the parametrix-based poten- s– tials the arguments from the proof of Lemma 4.1 in [11]. Let ψ ∈ H (∂). From (3.53) we obtain, in the sense of distributions, 1 ∇a ∗ ∗ ∗ AV ψ =–A P γ ψ =–γ ψ + ∇· P γ ψ a a ∗ n =–γ ψ – ∇· (∇a)V ψ in R , (3.81) ∗ ∗ wherewehavetaken into accountthat P γ ψ = γ ψ. Then, since the operator A is formally self-adjoint, for any test function φ ∈ D(R ), we obtain V ψ(y)Aφ(y) dy = AV ψ, φ n =–ψ, γφ – ∇· (∇a)V ψ , φ . (3.82) R ∂ n s– 1 3 2 n Note that, for a ∈ C (R )and ψ ∈ H (∂)with < s < , the continuity of operator 2 2 s n 2 n (3.68)and Theorem 2.6 imply that V ψ ∈ H (R )and (∇a)V ψ ∈ H (R )for some  ∈ loc loc (0, 1). Hence, from the second Green identity (2.21)with v = V ψ and u = φ,along with (3.81), we have V ψ(y)Aφ(y) dy – A V ψ, φ ± ± = V ψ(y)Aφ(y) dy – E r AV ψ, φ ± ± ± = V ψ(y)Aφ(y) dy + E r ∇· (∇a)V ψ , φ ± ± ± ± ± ± = ± T φ, γ V ψ ∓ T V ψ, γ φ . (3.83) ∂ ∂ ∗ ∗ Here we employed that r γ ψ =0 since supp γ ψ ⊂ ∂. Let us take into account + – + – c + that γ φ = γ φ = γφ and T φ = T φ = T φ due to smoothness of φ,whereas γ V ψ = Mikhailov Boundary Value Problems (2018) 2018:87 Page 18 of 52 γ V ψ = γ V ψ by the first relation in (3.79). Moreover, we also have ˚ ˚ E r ∇· (∇a)V ψ , φ = r ∇· (∇a)V ψ , E φ ± ± ± ± ± ± = ± (∂ a)γ V ψ, γφ – (∇a)V ψ, ∇φ . Then summing up (3.83)for  and  ,weobtain + – + – V ψ(y)Aφ(y) dy =– T V ψ – T V ψ, γφ + (∇a)V ψ, ∇φ . (3.84) ∂ R + – V ψ – T V ψ, γφ = ψ, γφ for arbitrary Comparing (3.84)and (3.82), we obtain T ∂ ∂ φ ∈ D(R ), which implies the first jump relation in (3.80). s– Let us similarly prove the second jump relation in (3.80). Let ϕ ∈ H (∂). From (3.54) we obtain, in the sense of distributions, 1 ∇a c∗ c∗ c∗ AW ϕ =–A P T ϕ =–T ϕ + ∇· P T ϕ a a c∗ n =–T ϕ – ∇· (∇a)W ϕ in R , (3.85) c∗ c∗ wherewehavetaken into accountthat P T ϕ = T ϕ. Then for any test function φ ∈ D(R ), we obtain c∗ W ϕ(y)Aφ(y) dy = AW ϕ, φ n =– T ϕ + ∇· (∇a)W ϕ , φ =– ϕ, T φ + (∇a)W ϕ, ∇φ . (3.86) ∂ R 1 3 2 n s– Note that for a ∈ C (R )and ϕ ∈ H (∂)with < s < , the continuity of operators 2 2 s s (3.69)and (3.70)and Theorem 2.6 imply that r W ϕ ∈ H ( ), r W ϕ ∈ H ( ), and +  – + – loc 1 + + 2 (∇a)r W ϕ ∈ H ( ), (∇a)r W ϕ ∈ H ( )forsome  ∈ (0, 1). Hence from the sec- +  – + – loc ond Green identity (2.21)for v = W ϕ and u = φ,along with (3.85), we have W ϕ(y)Aφ(y) dy – A W ϕ, φ ± ± = W ϕ(y)Aφ(y) dy – E r AW ϕ, φ ± ± ± = W ϕ(y)Aφ(y) dy + E r ∇· (∇a)W ϕ , φ ± ± ± ± ± ± = ± T φ, γ W ϕ ∓ T W ϕ, γ φ . (3.87) ∂ ∂ c∗ c∗ Here we employed that r T ϕ =0 since supp T ϕ ⊂ ∂. Let us also take into account + – + – c + that γ φ = γ φ = γφ and T φ = T φ = T φ due to smoothness of φ,whereas γ W ϕ – γ W ϕ =–ϕ by the second relation in (3.79). Moreover, we also have ˚ ˚ E r ∇· (∇a)W ϕ , φ = r ∇· (∇a)W ϕ , E φ ± ± ± ± ± ± = ± (∂ a)γ W ϕ, γφ – (∇a)W ϕ, ∇φ . ± Mikhailov Boundary Value Problems (2018) 2018:87 Page 19 of 52 Then summing up (3.87)for  and  ,weobtain + – W ϕ(y)Aφ(y) dy – (∂ a)ϕ, γφ – (∇a)W ϕ, ∇φ ∂ R c + – =– T φ, ϕ – T W ϕ – T W ϕ, γφ . (3.88) ∂ ∂ + – Comparing (3.88)and (3.86), we obtain T W ϕ – T W ϕ, γφ = (∂ a)ϕ, γφ for ar- ∂ ν ∂ bitrary φ ∈ D(R ), which implies the second jump relation in (3.80). Theorem 3.5(iii) and the first relation in (3.79) imply the following assertion. 1 3 Corollary 3.7 Let ∂ be a compact Lipschitz boundary, and let < s < . The following 2 2 operators are continuous: 3 1 + – s– s– s 2 2 V := γ V = γ V : H (∂) → H (∂), a ∈ C ( ); (3.89) 1 1 1 + – s– s– s 2 2 W := γ W + γ W : H (∂) → H (∂), a ∈ C ( ); (3.90) 1 3 3 + – s– s– 2 2 2 W := T V + T V : H (∂) → H (∂), a ∈ C ( ); (3.91) + ± 1 1 3 + – s– s– 2 2 2 L := T W + T W : H (∂) → H (∂), a ∈ C ( ). (3.92) For the case of smooth boundary, the boundary operators defined in Corollary 3.7 (see [27, Eq. (7.3)] for the fundamental solution-based potentials on Lipschitz domains) cor- respond to the boundary integral (pseudo-differential) operators of direct surface values W, and the co-normal derivatives of the single-layer potential, the double-layer potential of the single-layer potential W and of the double-layer potential (see [3, Eq. (3.6)-(3.8)]) for the parametrix-based potentials on smooth domains. See also [27, Theorems 7.3, 7.4] about integral representations on Lipschitz domains of the boundary operators associated with the layer potentials based on fundamental solutions. If a = 1, then we equip the operators defined in Corollary 3.7 with subscript .Then, under the hypotheses of Corollary 3.7,wehave(see[3, Eq. (3.10)–(3.13)] for the potentials on smooth domains) 1 1 Vg = V g, Wg = W (ag), (3.93) a a ∂ a ∂ a ν ν W g = W g – V g, Lg = L (ag)– W (ag). (3.94) a a Indeed, relations (3.93) immediately follow from (3.89), (3.90), and (3.56). Further, T Vg = + 1 T ( V g). Let {v }⊂ D()be a sequence such that v – V g 1 → 0as k →∞, k k s,– H (; which implies that also v – Vg → 0as k →∞.Then(see[32, Lemma 6.10]) s,– H (;A) 1 1 ∂ a + c c + T Vg = lim T v = lim aT v = lim ∂ v – γ v k k ν k k k→∞ a k→∞ a k→∞ a ∂ a + + = T V g – γ V g. a Mikhailov Boundary Value Problems (2018) 2018:87 Page 20 of 52 ∂ a – – ν – Similarly, T Vg = T V g – γ V g,which,togetherwith(3.91), implies the first relation in (3.94). The second relation in (3.94) is proved by similar arguments. Employing definitions (3.89)–(3.92), the jump properties (3.79)–(3.80)can be re-written 3 1 s– s– 1 3 2 2 for ψ ∈ H (∂)and ϕ ∈ H (∂)with < s < as follows: 2 2 ± ± s n γ V ψ = V ψ, γ W ϕ = ∓ ϕ + W ϕ if a ∈ C R ; (3.95) 1 1 ±  ± n T V ψ = ± ψ + W ψ, T W ϕ = ± (∂ a)ϕ + Lϕ if a ∈ C R . (3.96) ν + 2 2 4 The third Green identity and integral relations In this section, we apply some limiting procedures to obtain the parametrix-based third Green identity. 1 3 s s Theorem 4.1 Let  be a bounded Lipschitz domain, u ∈ H (), < s < , and a ∈ C (). 2 2 (i) The following generalised third Green identity holds: u + Ru + W γ u = PA u in , (4.1) where, by (2.9) and (2.10), ˇ ˇ ˇ ˚ PA u(y):= A u, P(·, y) =–E u, P(·, y) =– E (a∇u), ∇P(·, y) = ∇· P E (a∇u)(y), a.e. y ∈ , (4.2) a(y) and, particularly, if s =1, then PA u(y)=– a(x)∇u(x) ·∇ P(x, y) dx, a.e. y ∈ . (4.3) s–2 ˜ ˜ (ii) Moreover, if Au = r f in , where f ∈ H (), then the generalised third Green identity takes the form + + ˜ ˜ u + Ru – VT (f ; u)+ W γ u = Pf in . (4.4) Proof (i) Let first u ∈ D(). For y ∈ ,let B (y) ⊂  be a ball centred in y with sufficiently small radius ,and let  :=  \ B (y). For any fixed y, evidently, P(·, y)= P (·, y) ∈ a(y) 1,0 D( ) ⊂ H (A;  ) and has the coinciding classical and canonical co-normal derivatives on ∂ . Then from the first Green identity (2.20) applied to  with v = P(·, y)we obtain + + + + – T P(·, y), γ u – T P(·, y), γ u + R(·, y), u ∂B (y) ∂ =– ∇P(·, y), a∇u . (4.5) Since + + lim T P(·, y), γ u ∂B (y) →0 = lim ∂ P (x, y) a(x)γ u(x) dS(x)=–u(y), ν(x) →0 a(y) ∂B (y) Mikhailov Boundary Value Problems (2018) 2018:87 Page 21 of 52 by passing to the limits as  → 0equation (4.5) reduces to the third Green identity (4.1) for any u ∈ D(). Taking into account the denseness of D()in H () and the mapping properties of the volume potentials (3.30)and (3.35)inTheorem 3.2 and of the double- layer potential (3.69)inTheorem 3.5(i), we obtain that (4.1)–(4.2) also hold for any u ∈ s 1 3 s H ()with < s < in the sense of H (), which also implies (4.3)for s =1. 2 2 (ii) Let {u }∈ D() be a sequence converging to u in H (). By (4.2), (4.3), and (2.18) we have ˇ ˇ PA u (y)=– lim a(x)∇u (x) ·∇ P(x, y) dx =– lim E u , P(·, y) k k x  k →0 →0 = lim (A u )(x)P(x, y) dx – P(x, y)T u (x) dS(x) k k →0 ∂B (y) + + – P(x, y)T u (x) dS(x) = PA u (y)+ VT u (y). (4.6) k  k k s–2 s–2 s–2 s–2 ˜ ˜ ˇ ˜ Let now f := E r (A u – A u)+ f ,where E : H () → H ()is a (non-unique) k   k ˇ ˜ continuous extension operator, which exists by [31, Theorem 2.16]. Since r A u = r f , ˜ ˜ ˇ we obtain r f = r A u = r A u .Hence k   k   k s–2 ˜ ˇ ˇ r A u – r A u = r A (u – u) →0in H (), k     k s–2 ˜ ˜ and f → f in H ()as k →∞.Thenby (4.6), (3.53), and (2.19)weobtain + + –1 ˇ ˜ ˜ ˜ ˜ ˜ PA u = PA u + VT u = PA u + VT (f ; u )– V γ (f – A u ) k  k k  k k k k  k + + ˜ ˜ ˜ ˜ ˜ ˜ = PA u + VT (f ; u )+ P(f – A u )= VT (f ; u )+ Pf , k k k k  k k k k ∗ –1 ∗ ˜ ˜ ˜ ˜ wherewetookintoaccountthat γ (γ ) (f – A u )= f – A u by [31, Corollary 2.11] k  k k  k s–2 ˜ ˜ ˇ ˜ since f – A u ∈ H . Passing to the limits as k →∞,weobtain PA u(y)= Pf + k  k VT (f ; u), which by substitution into (4.1)gives (4.4). For some functions f , , , let us consider a more general “indirect” integral relation associated with (4.4): u + Ru – V  + W  = Pf in . (4.7) The following lemma extends Lemma 4.1 from [3], where the corresponding assertion was proved for f ∈ L (), s =1, a ∈ C (), and the infinitely smooth boundary. 3 1 1 3 s s s– s– 2 2 Lemma 4.2 Let < s < and a ∈ C (). Let u ∈ H (),  ∈ H (∂),  ∈ H (∂), 2 2 s–2 and f ∈ H () satisfy (4.7). Then Au = r fin , (4.8) + + V  – T (f ; u) – W  – γ u =0 in . (4.9) Proof Subtracting (4.7) from identity (4.1), we obtain ˇ ˜ V  – W  – γ u = P[A u – f]in . (4.10) Mikhailov Boundary Value Problems (2018) 2018:87 Page 22 of 52 Multiplying equality (4.10)by a, applying the Laplace operator , and taking into account ˜ ˇ ˜ (3.57)and (3.17), we get r f = r (A u)= Au in .Thismeans that f is an extension of the s–2 s–2 distribution Au ∈ H ()to H (), and u satisfies (4.8). Then (2.15)implies ˇ ˜ ˇ ˜ ˜ P[A u – f ](y)= A u – f , P(·, y) =– T (f ; u), P(·, y) = VT (f ; u)(y), y ∈ . (4.11) Substituting (4.11)into(4.10)leads to (4.9). 1 3 s s–1 ∇ For < s < , a ∈ C (), and g ∈ H (), let us introduce the operator A as 2 2 + A g := –∇· E (g∇a). (4.12) 1 3 Lemma 4.3 Let < s < . 2 2 |s–1|+1 (i) If a ∈ C (), then the following operator is continuous: ∇ s–1 s–2 A : H () → H (). (4.13) (ii) If a ∈ C (), then the following operator is continuous and compact: ∇ s s–2 A : H () → H (). (4.14) |s–1|+1 |s–1| Proof (i) If a ∈ C (), then ∇a ∈ C (), and by Theorem 2.6, ∇a is a multiplier in + + s–1 H (), which implies the continuity of operator (4.13). (ii) For 1 ≤ s < ,wehave s = |s –1| + 1, which by item (i) implies the continuity of operator (4.13) and thus the continuity and compactness of operator (4.14). For < s < 1, we need an estimate of the norm g∇a s–1 .First,byDefinition 2.5 the H () s 0,t t inclusion a ∈ C () implies that there exists t ∈ (s,1) such that a ∈ C ()= B ()= + ∞,∞ t t–1 F () (see, e.g., Proposition in [45, Sect. 2.1.2]) and hence ∇a ∈ F (). Then, by ∞,∞ ∞,∞ Theorem 1 from [45, Sect. 4.4.3], g∇a ≤ C ∇a g σ t–1 t–1 H () F () F () ∞,∞ 2,∞ ≤ C a g σ , ∀ σ ∈ (1 – t, s). (4.15) 0,t H () C () On the other hand, by (3.49), item (ii) of Proposition from [45, Sect. 2.2.1], and (4.15)we obtain A g ≤ c g∇a s–1 = c g∇a s–1 s–2 3 H () 3 F () H () 2,2 ≤ C g∇a t–1 ≤ C C a 0,t g . 1 1 H () F () C () 2,∞ ∇ σ s–2 Thus the operator A : H () → H () is continuous, which implies the continuity and, by the Rellich compact embedding theorem, also the compactness of operator (4.14)for < s <1. 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 23 of 52 In accordance with notation (2.10), let us also denote ˇ ˚ g := ∇· E r ∇g. Let us now discuss the trace and two forms of the co-normal derivative associated with equation (4.7). Lemma 4.4 (i) Under the hypotheses of Lemma 4.2, + + + γ u + γ Ru – V  –  + W  = γ Pf on ∂, (4.16) + + ∇ T (f ; u)+ T A u; aRu –  – W  + L (a) ˜ ˜ = T (f ; P f ) on ∂. (4.17) (ii) If, moreover, a ∈ C (), then + +  + + ˜ ˜ ˚ ˜ ˜ T (f ; u)+ T Ru –  – W  + T W  = T (f + ER f ; Pf ) on ∂, (4.18) where R is defined in (3.14) and (3.16). Proof (i) Equation (4.16)isimplied by (4.7)and (3.95). To prove (4.17), let us first multiply (4.7)by a to obtain –V  + W (a)= P f – au – aRu in . (4.19) Since {–V  +W (a)} = 0, for the both sides of (4.19) the canonical co-normal deriva- tive T is well defined, + + + –T V  + T W (a)= T (P f – au – aRu), (4.20) and by (2.17) + –1 ˜ ˜ T (P f – au – aRu)=– γ (P f – au – aRu), (4.21) because by (4.19), ˜ ˜ ˚ ˜ ˚ (P f – au – aRu)= E (P f – au – aRu)= E – V  + W (a) =0. Note that, by the second equality in (3.16), (aRu)=–∇· P (u∇a) =–∇· (u∇a)= r A u in , (4.22) ∇ s–2 s–2 which implies that A u ∈ H () is an extension of (aRu) ∈ H (). Further (see (2.10)), ˇ ˚ ˚ ˚ (au)= ∇· E r ∇(ua)= ∇· E r (u∇a)+ ∇· E r (a∇u) Mikhailov Boundary Value Problems (2018) 2018:87 Page 24 of 52 ∇ n =–A u + A u in R . Then ∇ n ˇ ˜ ˜ ˇ ˜ ˜ ˇ ˇ (P f – au – aRu)= f – P f –(f – A u)– A u – (aRu) in R , and by (2.13) + + + + ∇ ˜ ˜ ˜ ˜ T (P f – au – aRu)= T (f ; P f )– T (f ; u)– T A u; aRu . Substituting this in (4.20), we obtain + + ∇ + + + ˜ ˜ ˜ T (f ; u)+ T A u; aRu – T V  + T W (a)= T (f ; P f)on ∂. Taking into account jump relation (3.63)and (3.66)with(3.67), we arrive at (4.17). (ii) To prove (4.18), letusfirstremarkthat ˜ ˜ ˜ APf = f + R f in , (4.23) ˜ ˜ which implies, due to (4.8), that A(Pf – u)= R f in ,where R is defined in (3.14)and ∗ ∗ 2 σ 1 (3.16), and since a ∈ C (), we obtain by (3.39)that R f ∈ H ()for some σ >– .Then + ∗ σ s–2 ˜ ˜ ˜ ˚ ˜ A(Pf – u) can be canonically extended to A(Pf – u)= E R f ∈ H () ⊂ H (). This implies that there exists a canonical co-normal derivative of (Pf – u), for which, due to (2.17)and (2.13), we have + –1 ˜ ˜ ˜ ˇ ˜ ˇ T (Pf – u)= γ A(Pf – u)– A Pf + A u –1 ˚ ˜ ˇ ˜ ˇ = γ [E R f – A Pf + A u] –1 ˜ ˇ ˜ ˇ ˜ ˚ ˜ = γ [f + E R f – A Pf + A u – f ] + + ˜ ˜ ˜ ˜ = T (f + E R f , Pf )– T (f , u), (4.24) s–2 ˜ ˚ ˜  ˜ where f + E R f ∈ H () is an extension of APf due to (4.23). From (4.7)wehave Pf – u = Ru – V  + W  in . Substituting this in the left-hand side of (4.24) and taking into account jump relation (3.96), we arrive at (4.18). Note that, unlike (4.17), the co-normal derivative form (4.18)ofrelation(4.7)iswritten without referring to the corresponding constant-coefficient potentials. 1 3 –1/2 s–2 Remark 4.5 Let < s < and f ∈ H () ⊂ H (). 2 2 s,–1/2 (i) Then evidently P f ∈ H (, ) and + + ˜ ˜ ˜ T (f ; P f )= T P f . (4.25) –1/2 (ii) Furthermore, if the hypotheses of Lemma 4.2 aresatisfiedand f ∈ H (),then s,–1/2 + + + ˜ ˜ (4.8)implies that u ∈ H (, A) and T (f ; u)= T (Au; u)= T u.Henceforth, (4.17) takes the simpler form + + ∇  + + T u + T A u; aRu –  – W  + L (a)= T P f on ∂. (4.26) 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 25 of 52 s,–1/2 If, in addition, au ∈ H (, ),thenby (4.22) ∇ 2 (aRu)= r A u =–∇· (u∇a)= Au – (au) ∈ H (). Hence the canonical co-normal derivative T (aRu) is well defined, and by (2.13), (2.17), (3.16), and (4.22) + ∇ T A u; aRu –1 ∇ = γ A u – (aRu) –1 ∇ + = γ A u – (aRu) + T (aRu) –1 + ˚ ˚ = γ –∇· E (u∇a)+ E ∇· (u∇a) + T (aRu) ∗ ∗ –1 –1 + ˇ ˜ ˇ ˜ = γ [Au – Au]+ γ – (au)+ (au) + T (aRu) + + + =–T u + T (au)+ T (aRu). (4.27) This reduces (4.26)tothe relation + +  + + T (au)+ T (aRu)–  – W  + L (a)= T P f on ∂ (4.28) with only canonical normal derivatives associated with the Laplace operator involved. –1/2 (iii) If the hypotheses of Lemma 4.2 aresatisfiedand,moreover, f ∈ H (),and 2 s,– –1/2 ˜ ˜ a ∈ C (), then, by (3.32)and (3.39), Pf ∈ H (; A) and R f ∈ H (), + ∗ + + ˜ ˚ ˜ ˜ ˜ implying T (f + E R f ; Pf )= T (Pf ).Henceforth, (4.18) reduces to the relation + +  + + T u + T Ru –  – W  + T W  = T Pf on ∂ with only canonical co-normal derivatives associated with the operator A involved. Remark 4.6 (i) Let the hypotheses of Lemma 4.2 be satisfied and suppose that a sequence – s–2 ˜ ˜ {f }∈ H ()converges to f in H (). By the continuity of operators (3.30)and (3.34), estimate (2.14), and relation (4.25)for f ,weobtainthat + + s– ˜ ˜ ˜ T (f ; P f )= lim T P f in H (∂); j→∞ see also Theorem 7.1. (ii) If, moreover, a ∈ C (), then, similarly, + + + ˜ ˚ ˜ ˜ ˜ ˚ ˜ ˜ ˜ T (f + E R f , Pf )= lim T (f + E R f , Pf )= lim T Pf . ∗ j  ∗ j j j j→∞ j→∞ Lemma 4.4 and the third Green identity (4.4) imply the following assertion. 1 3 s Corollary 4.7 Let  be a bounded Lipschitz domain, and let < s < , a ∈ C (), u ∈ 2 2 + s s–2 ˜  ˜ H (), and f ∈ H () be such that Au = r fin . Mikhailov Boundary Value Problems (2018) 2018:87 Page 26 of 52 (i) Then + + + + + ˜ ˜ γ u + γ Ru – VT (f ; u)+ W γ u = γ Pf on ∂, (4.29) + + ∇  + + ˜ ˜ T (f , u)+ T A u; aRu – W T (f , u)+ L aγ u ˜ ˜ = T (f ; P f ) on ∂. (4.30) (ii) If, moreover, a ∈ C (), then + +  + + + ˜ ˜ T (f , u)+ T Ru – W T (f , u)+ T W γ u ˜ ˚ ˜ ˜ = T (f + E R f , Pf ) on ∂, (4.31) where R is defined in (3.14) in (3.16). 1 3 Let us extend to Lipschitz domains and s ∈ ( , ) Lemma 4.2(i,ii) from [3], which is 2 2 proved there for smooth domains and s =1. Lemma 4.8 Let  be a bounded simply connected Lipschitz domain, and let a ∈ C () 1 3 with < s < . 2 2 ∗ s– ∗ ∗ (i) If  ∈ H (∂) and r V  =0, then  =0. ∗ s– ∗ ∗ (ii) If  ∈ H (∂) and r W  =0, then  =0. Proof To prove (i), let us multiply equation r V  =0 by a,which by thefirstrelationin (3.56)reduces it to r V  =0 in . Taking the trace of this equation on ∂ and using the first relation in (3.95)(forthe case a =1), by Theorem 7.3 we obtain item (i). Similarly, multiplying the equation r W  =0 by a, the second relation in (3.56)re- duces it to r W (a )=0 in . Taking the trace of this equation on ∂ and using the 1 ∗ ∗ ˆ ˆ first jump relation in (3.95)(forthe case a =1), we obtain –  + W  =0 on ∂,where ∗ ∗ ∗ ˆ ˆ = a .Since this equation for  is uniquely solvable (see Theorem 7.3), by condition (2.5) this implies item (ii). Theorem 4.9 Let  be a bounded simply connected Lipschitz domain, and let a ∈ C () 1 3 s–2 s ˜  ˜ with < s < . Let f ∈ H (). A function u ∈ H () is a solution of PDE Au = r fin  if 2 2 and only if it is a solution of boundary-domain integro-differential equation (4.4). Proof If u ∈ H ()solves PDE Au = r f in , then by Theorem 4.1(ii) it satisfies (4.4). On the other hand, if u solves the boundary-domain integro-differential equation (4.4), then + + using Lemma 4.2 for  = T (f ; u)and  = γ u completes the proof. 5 Segregated BDIE systems for the Dirichlet problem 1 3 For < s < , let us consider the Dirichlet problem: 2 2 Find a function u ∈ H () satisfying the equations Au = f in , (5.1) γ u = ϕ on ∂, (5.2) s–2 s– where f ∈ H () and ϕ ∈ H (∂). 0 Mikhailov Boundary Value Problems (2018) 2018:87 Page 27 of 52 Equation (5.1) is understood in the distributional sense (2.7), and the Dirichlet boundary condition (5.2) is understood in the trace sense. The following uniqueness assertion is well known for s = 1 and follows from the first Green identity; hence it also holds for 1 ≤ s < 3/2. |s–1| 3 Theorem 5.1 Let a ∈ C () with 1 ≤ s < . The Dirichlet problem (5.1)–(5.2) has at most one solution in H (). 5.1 BDIE formulations and equivalence to the Dirichlet problem 1 3 Let < s < . In this section, we reduce the Dirichlet problem (5.1)–(5.2) to three differ- 2 2 ent segregated boundary-domain integral equation (BDIE) systems. Two of these formu- lations, for s = 1 and infinitely smooth coefficients and infinitely smooth boundary, were analysed in [33]. s–2 s–2 ˜  ˜ Let f ∈ H () be an extension of f ∈ H ()(i.e., f = r f ), which always exists; see [31, Lemma 2.15 and Theorem 2.16]. Let us substitute into (4.4), (4.29), (4.30), and (4.31) the generalised co-normal derivative and the trace of the function u as + + T (f ; u)= ψ, γ u = ϕ , where ϕ is the known right-hand side of the Dirichlet boundary condition (5.2), and ψ ∈ s– H (∂) is a new unknown function that will be regarded as formally segregated from u. s s– Thus we will look for the unknown couple (u, ψ) ∈ H () × H (∂). BDIE system (D1). Let a ∈ C (). To reduce the Dirichlet BVP (5.1)–(5.2)tothe BDIE system (D1), we will use equation (4.4)in  and equation (4.29)on ∂. Then we arrive at the following system of the boundary-domain integral equations, (D1), which is similar to the corresponding system in [33]: D1 u + Ru – V ψ = F in , (5.3) + D1 γ Ru – V ψ = F on ∂, (5.4) where D1 D F F 1 0 D1 D F = = and F := Pf – W ϕ in . (5.5) D1 D F γ F – ϕ 2 0 s– s–2 D s Note that, for ϕ ∈ H (∂)and f ∈ H (), we have the inclusion F ∈ H ()due to the mapping properties of the Newtonian (volume) and layer potentials; see (3.30)and D1 s s– (3.69). Hence F ∈ H () × H (∂). BDIE system (D2 ). Let a ∈ C (). To obtain a segregated BDIE system of the second kind,wewilluse equation (4.4)in  and equation (4.30)on ∂. Then we arrive at the following BDIE system (D2 ): D2 u + Ru – V ψ = F in , (5.6) + ∇  D2 ψ + T A u; aRu – W ψ = F on ∂, (5.7) 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 28 of 52 where D2 F Pf – W ϕ D2 F = = . (5.8) D2 ˜ ˜ F T (f ; P f )– L (aϕ ) D2 Due to the mapping properties of the operators involved in (5.11), we have F ∈ s s– H () × H (∂). BDIE system (D2). Let the coefficient be smoother than in the first two cases, a ∈ C (). Now we will use equation (4.4)in  and equation (4.31)on ∂. Then we arrive at another BDIE system of the second kind, (D2), which is similar to the corresponding system in [33]: D2 u + Ru – V ψ = F in , (5.9) +  D2 ψ + T Ru – W ψ = F on ∂, (5.10) where D2 F Pf – W ϕ D2 1 F = = . (5.11) D2 + + ˜ ˚ ˜ ˜ F T (f + E r R f ; Pf )– T W ϕ ∗ 0 D2 s Due to the mapping properties of the operators involved in (5.11), we have F ∈ H () × s– H (∂). LetusprovethatBVP(5.1)–(5.2)in  is equivalent to each of the three systems of BDIEs, (D1), (D2 ), and (D2). s 1 3 s– s–2 Theorem 5.2 Let a ∈ C () with < s < . Let ϕ ∈ H (∂), f ∈ H (), and f ∈ 2 2 s–2 H () be such that r f = f . (i) If a function u ∈ H () solves the Dirichlet BVP (5.1)–(5.2), then the couple s s– (u, ψ) ∈ H () × H (∂), where ψ = T (f ; u) on ∂ (5.12) solves the BDIE systems (D1), (D2 ) and, if a ∈ C (), then also the BDIE system (D2). s s– (ii) Vice versa, if a a couple (u, ψ) ∈ H () × H (∂) solves one of the BDIE systems, (D1), (D2 ), or (D2) (if a ∈ C ()), then this solution solves the other BDIE systems, whereas u solves the Dirichlet BVP, and ψ satisfies (5.12). Proof (i) Let u ∈ H ()be a solution to BVP (5.1)–(5.2). Setting ψ by (5.12) evidently im- s– plies ψ ∈ H (∂). Then it immediately follows from Theorem 4.9 and relations (4.29)– (4.31) that the couple (u, ψ)solves systems (D1), (D2) ,and,if a ∈ C (), then also (D2), with the right-hand sides (5.5), (5.8), and (5.11), respectively, which completes the proof of item (i). s s– (ii) Let now a couple (u, ψ) ∈ H () × H (∂)solve BDIE system (5.3)–(5.4). Taking the trace of equation (5.3)on ∂, and subtracting equation (5.4)fromit, we obtain γ u = ϕ on ∂, (5.13) 0 Mikhailov Boundary Value Problems (2018) 2018:87 Page 29 of 52 i.e., u satisfies the Dirichlet condition (5.2). Equation (5.3) and Lemma 4.2 with  = ψ and = ϕ imply that u is a solution of PDE (5.1), and ∗ ∗ V  – W  =0 in , ∗ + ∗ + ∗ where  = ψ – T (f ; u)and  = ϕ – γ u.Due to equation (5.13),  =0. Then Lemma 4.8(i) implies  = 0, i.e., condition (5.12). Thus u obtained from solution of BDIE system (D1) solves the Dirichlet problem and hence, by item (i) of the theorem, (u, ψ) solves also BDIE system (D2 )and,if a ∈ C (), then also (D2). s s– Let now a couple (u, ψ) ∈ H () × H (∂)solve BDIE system (5.6)–(5.7). Lemma 4.2 for equation (5.6)implies that u is a solution of PDE (5.1), and equation (4.9)holds for = ψ and  = ϕ , whereas Corollary 4.7 gives equation (4.30). Multiplication of (4.9)by a reduces it to + + V ψ – T (f ; u) – W a ϕ – γ u =0 in . (5.14) Subtracting (4.30)from equation(5.7) and taking into account (5.14)give ψ – T (f ; u)=0 on ∂, (5.15) that is, equation (5.12)isproved. Equations(5.14)and (5.15)give W  =0 in ,where ∗ + ∗ = a(ϕ – γ u). Then Lemma 4.8(ii) implies  =0 on ∂.Thismeans that u satisfies the Dirichlet condition (5.2). Thus u obtained from solution of BDIE system (D2 )solves the Dirichlet problem, and hence, by item (i) of the theorem, the couple (u, ψ)solves also BDIE system (D1) and, if a ∈ C (), then also (D2). 2 s s– Let, finally, a ∈ C (), and let a couple (u, ψ) ∈ H () × H (∂)solve BDIE system (5.9)–(5.10). Lemma 4.2 for equation (5.9)implies that u is a solution of PDE (5.1), and equation (4.9)holds for  = ψ and  = ϕ , whereas Corollary 4.7 gives equation (4.31). Subtracting (4.31)fromequation(5.10) and adding to it the canonical co-normal deriva- + ∗ tive T of equation (4.9)leadto(5.12). Equations (4.9)and (5.12)imply W  =0 in , ∗ + ∗ where  = ϕ – γ u. Then by Lemma 4.8(ii) we deduce  =0 on ∂.Thismeans that u satisfies the Dirichlet condition (5.2). Thus u obtained from solution of BDIE system (D2) solves the Dirichlet problem, and hence, by item (i) of the theorem, the couple (u, ψ) solves also BDIE systems (D1) and (D2 ). 5.2 Properties of BDIE system operators for the Dirichlet problem BDIE systems (D1), (D2 ), and (D2) can be written as 1 D D1 2 D D2 2 D D2 D U = F , D U = F ,and D U = F , D  s s– respectively. Here U := (u, ψ) ∈ H () × H (∂), I – R –V D := , γ R –V I + R –V I + R –V D := , D := , 1 1 + ∇  + T (A ; aR) I – W T R I – W 2 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 30 of 52 D1 D2 D2 whereas F , F ,and F are given by (5.5), (5.8), and (5.11), respectively. Note that + ∇ –1 ∇ T A ; aR u := γ A u – (aRu) . (5.16) 1 3 Let < s < . The operators 2 2 3 1 1 s s– s s– s 2 2 D : H () × H (∂) → H () × H (∂)if a ∈ C (), (5.17) 1 3 s s– s s– s 2 2 D : H () × H (∂) → H () × H (∂)if a ∈ C (), (5.18) 1 3 2 s s– s s– 2 2 2 D : H () × H (∂) → H () × H (∂)if a ∈ C (), (5.19) are continuous due to the mapping properties of the operators constituting them (see Sect. 3), whereas for the right-hand sides of the BDIE systems, we have the inclusions 1 3 3 D1 s s– D2 s s– D2 s s– 2 2 2 F ∈ H () × H (∂), F ∈ H () × H (∂), and F ∈ H () × H (∂). 1 3 Theorem 5.3 Let  be a bounded simply connected Lipschitz domain, and let < s < . 2 2 Operators (5.17)–(5.19) are Fredholm operators with zero index. Proof The continuity of operators has been already proved. To prove the Fredholm property of operator (5.17), let us consider the operator 3 1 I –V 1 s s– s s– 2 2 D := : H () × H (∂) → H () × H (∂). (5.20) 0–V As a result of compactness properties of the operators R and γ R given by (3.35)and (3.42)inTheorem 3.2,operator(5.20) is a compact perturbation of operator (5.17). The operator D is an upper triangular matrix operator with the following scalar diagonal in- vertible operators: s s I : H () → H (), 3 1 s– s– 2 2 V : H (∂) → H (∂), where the invertibility of the operator V is implied by the invertibility of operator V in (7.4)and by thefirstrelationin(3.93). This implies that operator (5.20) is invertible. Thus (5.17) is a Fredholm operator with zero index. The operator I –V 3 3 2 s s– s s– 2 2 D := : H () × H (∂) → H () × H (∂) (5.21) 0 I – W s s is a compact perturbation of operator (5.18). Indeed, the operators R : H () → H () + ∇ s is compact due to Theorem 3.2. The compactness of the operator T (A ; aR): H () → s– ∇ s s–2 H (∂), defined by (5.16), follows from that of the operator A : H () → H ()given s s by Lemma 4.3(ii) and of the operator R : H () → H (), i.e., operator (3.35)inTheo- rem 3.2. Mikhailov Boundary Value Problems (2018) 2018:87 Page 31 of 52 Consider the diagonal operators of the upper triangular matrix operator D .The op- s s erator I : H () → H () is evidently invertible, whereas the invertibility of the operator 3 3 s– s– 2 2 I – W : H (∂) → H (∂)is stated by Theorem 7.3. This implies that operator (5.21) is invertible, and hence operator (5.18) is Fredholm with zero index. Operator (5.21)isalsoacompactperturbationofoperator(5.19). Indeed, the opera- s s + s s– tors R : H () → H ()and T R : H () → H (∂) are compact due to Theorem 3.2. 2  ∂ a From the first representation in (3.94), for a ∈ C (), the operator W – W = V : s– σ 1 1 H (∂) → H (∂), where σ = min{ , s – }, is continuous, which implies that the op- 2 2 3 3 s– s– 2 2 erator W – W : H (∂) → H (∂) is compact. Since operator (5.21) is invertible, this implies that operator (5.19) is Fredholm with zero index. 1 3 Theorem 5.4 Let  be a bounded simply connected Lipschitz domain, < s < , and σ = 2 2 max{1, s}. The following operators are continuously invertible: 3 1 1 s s– s s– σ 2 2 D : H () × H (∂) → H () × H (∂) if a ∈ C (), (5.22) 1 3 s s– s s– σ 2 2 D : H () × H (∂) → H () × H (∂) if a ∈ C (), (5.23) 1 3 2 s s– s s– 2 2 D : H () × H (∂) → H () × H (∂) if a ∈ C (). (5.24) Proof First, let 1 ≤ s < .Then σ = s, and the injectivity of operators (5.22)–(5.24)isim- plied by the equivalence Theorem 5.2(ii) and the BVP uniqueness Theorem 5.1. Indeed, consider, for example, the injectivity of operator (5.22). For the homogeneous equation 1 D D1 D U = 0, its zero right-hand side F =0 can be represented as in (5.5)interms of f =0 D + and ϕ =0. Then, by Theorem 5.2(ii), U =(u, T (0; u)) ,where u is a solution of the Dirichlet problem (5.1)–(5.2) with the right-hand sides f =0 and ϕ = 0, which has only the trivial solution u =0 due to Theorem 5.1. The arguments for the injectivity of opera- tors (5.23)and (5.22)are similar. Since, by Theorem 5.3, operators (5.22)–(5.24) are Fredholm with zero index, this im- plies their invertibility for 1 ≤ s < . 1 2 Let now < s ≤ 1. Then σ = 1, i.e., a ∈ C () for operators (5.22)–(5.23), and a ∈ C () for operator (5.24). Hence, for a fixed function a satisfying the corresponding conditions in (5.22)–(5.23), all these operators are continuous for < s ≤ 1. By Theorem 5.3 they are also Fredholm with zero index. Since, as already proved, at s = 1, these operators are also invertible, Lemma 7.5 implies that their kernels (null-spaces) consist of only the zero element for any s ∈ ( , 1], which implies that the operators are invertible for all s from this interval. Theorems 5.4 and 5.2 imply the following assertion. 1 3 Corollary 5.5 Let  be a bounded simply connected Lipschitz domain, < s < , f ∈ 2 2 s–2 s– σ H (), ϕ ∈ H (∂), and a ∈ C () with σ = max{1, s}. Then the Dirichlet problem s D –1 (5.1)–(5.2) is uniquely solvable in H (). The solution is u =(A ) (f , ϕ ) , where the s– D –1 s–2 s inverse operator (A ) : H () × H (∂) → H () to the left-hand side operator s– D s s–2 A : H () → H () × H (∂) of the Dirichlet problem (5.1)–(5.2) is continuous. s–2 s–2 Remark 5.6 For a given function f ∈ H (), its extension f ∈ H () is not unique. Nevertheless, since the solution of the Dirichlet BVP (5.1)–(5.2) does not depend on this Mikhailov Boundary Value Problems (2018) 2018:87 Page 32 of 52 extension, equivalence Theorem 5.2(ii) implies that u in the solution of BDIE systems (D1) and (D2) does not depend on the particular choice of extension f although ψ obviously does; see (5.12). 6 Segregated BDIE systems for the Neumann problem Let us consider the Neumann problem: Find a function u ∈ H () satisfying the equations Au = r f in , (6.1) T (f ; u)= ψ on ∂, (6.2) s– s–2 where ψ ∈ H (∂)and f ∈ H (). Equation (6.1) in understood in the distribution sense (2.7), and the Neumann bound- ary condition (6.2)inthe sense(2.13). The following assertion is well known and can be proved, e.g., using variational settings and the Lax–Milgram lemma. Theorem 6.1 Let s =1 and a ∈ L (). (i) The homogeneous Neumann problem (6.1)–(6.2) admits only one linearly 0 1 independent solution u =1 in H (). (ii) The non-homogeneous Neumann problem (6.1)–(6.2) is solvable if and only if 0 + 0 f , u – ψ , γ u = 0. (6.3) |s–1| Remark 6.2 Item (i) in Theorem 6.1 evidently implies that, for 1 ≤ s < and a ∈ C (), the homogeneous Neumann problem associated with (6.1)–(6.2) also admits only one lin- 0 s early independent solution u =1 in H (). 6.1 BDIE formulations and equivalence to the Neumann problem 1 3 Let < s < . We will explore different possibilities of reducing the Neumann problem 2 2 (6.1)–(6.2) to a BDIE system. Let us represent in (4.4), (4.29), (4.30), and (4.31)the gener- alised co-normal derivative and the trace of the function u as + + T (f ; u)= ψ , γ u = ϕ, where ψ is the known right-hand side of the Neumann boundary condition (6.2), and s– ϕ ∈ H (∂) is a new unknown function that will be regarded as formally segregated s s– from u. Thus we will look for the unknown couple (u, ϕ) ∈ H () × H (∂). BDIE system (N1 ). Let a ∈ C (). Using equation (4.4)in  and equation (4.30)on ∂, we arrive at the following BDIE system (N1 ) of two equations for the couple of unknowns (u, ϕ): N1 u + Ru + W ϕ = F in , (6.4) + ∇ N1 T A u; aRu + L (aϕ)= F on ∂, (6.5) where N1 F Pf + V ψ N1 F = = . (6.6) N1 ˜ ˜ F T (f ; P f )– ψ + W ψ 0 0 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 33 of 52 N1 Due to the mapping properties of the operators involved in (6.9), we have F ∈ H () × s– H (∂). BDIE system (N1). Let the coefficient be smoother than in the previous case, a ∈ C (). Now, using equation (4.4)in  and equation (4.31)on ∂, we arrive at the following BDIE system (N1) of two equations for the couple of unknowns (u, ϕ), which is similar to the corresponding system in [33]: N1 u + Ru + W ϕ = F in , (6.7) + + N1 T Ru + T W ϕ = F on ∂, (6.8) where N1 F Pf + V ψ N1 1 F = = . (6.9) N1 + 1 ˜ ˜ ˜ F T (f + E r R f ; Pf )– ψ + W ψ ∗ 0 0 N1 s Due to the mapping properties of the operators involved in (6.9), we have F ∈ H () × s– H (∂). BDIE system (N2). Let again a ∈ C (). If we use equation (4.4)in  and equation (4.29) on ∂, we arrive for the couple (u, ϕ) at the following BDIE system (N2) of two equations of the second kind, which is also similar to the corresponding system in [33]: N2 u + Ru + W ϕ = F in , (6.10) + N2 ϕ + γ Ru + W ϕ = F ,on ∂. (6.11) where N2 N F F N2 1 0 N F = = , F := Pf + V ψ in . (6.12) N2 + N F γ F 2 0 N2 s Due to the mapping properties of the operators involved in (6.12), we have F ∈ H () × s– H (∂). 1 3 s s– s–2 Theorem 6.3 Let < s < , a ∈ C (), ψ ∈ H (∂), and f ∈ H (). 2 2 (i) If a function u ∈ H () solves the Neumann problem (6.1)–(6.2), then the couple + s– 2 (u, ϕ) with ϕ = γ u ∈ H (∂) solves BDIE systems (N1 ), (N2), and, if a ∈ C (), also (N1). s s– (ii) Vice versa, if a couple (u, ϕ) ∈ H () × H (∂) solves one of the BDIE systems, (N1 ), (N2), or (N1) (if a ∈ C ()), then the couple solves the other two BDE systems, whereas u solves the Neumann problem (6.1)–(6.2) and γ u = ϕ. Proof (i) Let u ∈ H () be a solution of the Neumann problem (6.1)–(6.2). Then from Theorem 4.9 and relations (4.29)–(4.31) we see that the couple (u, ϕ)with ϕ = γ u solves BDIE systems (N1 ), (N2), and (N1) with the right-hand sides (6.6), (6.12), and (6.9), re- spectively, which proves item (i). s s– (ii) Let a couple (u, ϕ) ∈ H () × H (∂)solve BDIE system (N1 ). Lemma 4.2 for equation (6.4)implies that u is a solution of PDE (6.1), and equation (4.9)holds for  = ψ 0 Mikhailov Boundary Value Problems (2018) 2018:87 Page 34 of 52 and  = ϕ, whereas Corollary 4.7 gives equation (4.31). Multiplication of (4.9)by a reduces it to + + V ψ – T (f ; u) – W a ϕ – γ u =0 in . (6.13) Subtracting (4.31)fromequation(6.5), we get T (f ; u)= ψ on ∂, i.e., u satisfies the Neu- mann condition (6.2). Further, from (6.13)wederive W (a(ϕ – γ u)) = 0 in ,whence γ u = ϕ on ∂ by Lemma 4.8, completing item (ii) for BDIE system (N1 ). Let a couple (u, ϕ) ∈ H () × H (∂) solve BDIE system (N1). Lemma 4.2 for equation (6.7)implies that u is a solution of PDE (6.1), and equation (4.9)holds for  = ψ and = ϕ, whereas Corollary 4.7 gives equation (4.31). Subtracting (4.31)from equation(6.8) gives T (f ; u)= ψ on ∂, i.e., u satisfies the Neumann condition (6.2). Further, from (4.9) + + we derive W(γ u – ϕ)=0 in ,whence γ u = ϕ on ∂ by Lemma 4.8,completingitem (ii) for BDIE system (N1). Let now a couple (u, ϕ) ∈ H () × H (∂) solve BDIE system (N2). Further, taking the trace of (6.10)on ∂ and comparing the result with (6.11), we easily derive that γ u = ϕ on ∂. Lemma 4.2 for equation (6.10)implies that u is a solution of PDE (6.1), and equations (4.9)holds for  = ψ and  = ϕ.Further,from(4.9) and relation γ u = ϕ we derive V ψ – T (f ; u) =0 in , whence T (f ; u)= ψ on ∂ by Lemma 4.8, i.e., u solves the Neumann problem (6.1)–(6.2), which completes the proof of item (ii) for BDIE system (N2). 6.2 Properties of BDIE system operators for the Neumann problem BDIE systems (N1 ), (N1), and (N2) can be written, respectively, as N N1 1 N N1 2 N N2 N U = F , N U = F , N U = F , N  s s– where U =(u, ϕ) ∈ H () × H (∂), I + R W I + R W N := , N := , + ∇ + + T (A ; aR) L T R T W I + R W N := , + 1 γ R I + W 1 3 and we denoted L g := L (ag). Let < s < . Due to the mapping properties of the poten- 2 2 tials (see Sect. 3), the operators 1 3 s s– s s– s 2 2 N : H () × H (∂) → H () × H (∂)if a ∈ C (), (6.14) 1 3 1 s s– s s– 2 2 2 N : H () × H (∂) → H () × H (∂)if a ∈ C (), (6.15) 1 1 2 s s– s s– s 2 2 N : H () × H (∂) → H () × H (∂)if a ∈ C () (6.16) are continuous, whereas for the right-hand sides of the BDIE systems, we have the inclu- 3 3 1 N1 s s– N1 s s– N2 s s– 2 2 2 sions F ∈ H () × H (∂), F ∈ H () × H (∂), F ∈ H () × H (∂). Mikhailov Boundary Value Problems (2018) 2018:87 Page 35 of 52 1 3 Theorem 6.4 Let  be a bounded simply connected Lipschitz domain, and let < s < . 2 2 Operators (6.14)–(6.16) are Fredholm operators with zero index. Proof The continuity of operators is already proved. Let us consider operator (6.14). Due to estimate (2.5)and Theorem 7.3, the operator L : 1 3 s– s– 2 2 H (∂) → H (∂) is a Fredholm operator with zero index. Therefore the operator IW 1 3 1 s s– s s– 2 2 N := : H () × H (∂) → H () × H (∂) (6.17) 0 L is also Fredholm with zero index. Operator (6.14)isacompactperturbationof N since the operators s s R : H () → H (), (6.18) + ∇ s s– T A ; aR : H () → H (∂) (6.19) are compact due to Theorem 3.2,ashas been showninthe compactness proofrelated to operator (5.21). Thus operator (6.14) is Fredholm with zero index as well. Operator (6.17) is also a compact perturbation of operator (6.15). Indeed, the operators (6.18), 1 3 + s– s– 2 2 T W – L : H (∂) → H (∂), + s s– T R : H () → H (∂) are compact, due to relations (3.94)and (3.96)and Theorem 3.7.Thusoperator(6.15)is Fredholm with zero index as well. To analyse operator (6.16), let us consider the auxiliary operator 1 1 IW 2 s s– s s– 2 2 N := : H () × H (∂) → H () × H (∂). (6.20) 0 I + W 1 1 1 For any function g,wecan represent ( I + W)g = ( I + W )(ag), which, by Theorem 7.3, 2 a 2 1 1 1 s– s– 2 2 implies that the operator I + W : H (∂) → H (∂) and hence operator (6.20)are Fredholm with zero index. Due to the compactness of operator (6.18), operator (6.16)isa compact perturbation of operator (6.20) and thus is Fredholm with zero index as well. 1 3 Theorem 6.5 Let  be a bounded simply connected Lipschitz domain, < s < , and σ = 2 2 max{1, s}. The following operators have one-dimensional null-spaces, ker N = ker N = s– 2 s 0 0 ker N , in H () × H (∂), spanned over the element (u , ϕ )=(1,1): 1 3 s s– s s– σ 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (), (6.21) 1 3 1 s s– s s– 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (), (6.22) 1 1 2 s s– s s– σ 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (). (6.23) + Mikhailov Boundary Value Problems (2018) 2018:87 Page 36 of 52 Proof The conditions on the coefficient a imply that, for s = 1, operators (6.21)–(6.23)are continuous. Then the equivalence Theorem 6.3 and Theorem 6.1(i) imply that the homo- geneous BDIE systems (N1 ), (N1), and (N2) have only one linear independent solution 0 0 0   1 U =(u , ϕ ) = (1, 1) in H () × H (∂). Indeed, consider, for example, the homoge- N N1 neous equation N U = 0. Its zero right-hand side F =0 can be represented as in N + (6.6)interms of f =0 and ψ =0. Then, by Theorem 6.3(ii), U =(u, γ u) ,where u is a solution of the Neumann problem (6.1)–(6.2) with the right-hand sides f =0 and ψ =0, which has only the one linearly independent solution, u =1, due to Theorem 6.1.This proves the theorem for s = 1, and then Lemma 7.5 and Theorem 6.4 complete the proof 1 3 for < s < . 2 2 1 3 Lemma 6.6 Let  be a bounded simply connected Lipschitz domain, < s < , and 2 2 σ s s– a ∈ C () with σ = max{1, s}. For any couple (F , F ) ∈ H () × H (∂), there exists 1 2 s– s–2 auniquecouple (f ,  ) ∈ H () × H (∂) such that ∗ ∗ F = Pf – W  in , (6.24) 1 ∗ ∗ ˜ ˜ F = T (f ; Pf )– L (a ) on ∂. (6.25) 2 ∗ ∗ 3 1 s s– s–2 s– 2 2 Moreover,(f ,  )= C (F , F ), and C : H () × H (∂) → H () × H (∂) is a ∗ ∗ ∗ 1 2 ∗ linear continuous operator given by ˜ ˇ f = (aF )+ γ F , (6.26) ∗  1 2 –1 1 1 + ∗ = – I + W γ –aF + P (aF )+ γ F . (6.27) 1 2 a 2 s–2 s– Proof Let us first assume that there exist (f ,  ) ∈ H () × H (∂)satisfying equa- ∗ ∗ tions (6.24)and (6.25) and find their expressions in terms of F and F . Multiplying (6.24) 1 2 by a,weget aF – P f =–W (a )in . (6.28) Applying the Laplace operator to (6.28), we obtain ˜ ˜ (aF – P f )= (aF )– f =– W (a )=0 in , (6.29) ∗ 1 ∗ which means (aF )= r f in  (6.30) 1  ∗ s,0 + and aF – P f ∈ H (; ). Applying the canonical co-normal derivative operator T to ˜ ˜ ˜ both sides of equation (6.28) and taking into account that – W (a )= (aF – P f )= ∗ 1 0because W (a ) is a harmonic function in ,weobtain, dueto(2.17)and (2.13), –L (a )=–T W (a ) = T (aF – P f ) Mikhailov Boundary Value Problems (2018) 2018:87 Page 37 of 52 –1 ˜ ˜ ˜ ˇ = γ (aF – P f )– (aF – P f ) ∗  1 –1 + ˜ ˜ =– γ (aF – P f )= T (0; aF – P f ), (6.31) ∗ 1 where (6.30) was taken into account. Substituting this into (6.25), we obtain F = T (f , aF )on ∂. (6.32) 2 ∗ 1 Due to (6.30), we can represent ˜ ˇ ˜ ˚ f = (aF )+ f = ∇· E ∇(aF )– γ  , (6.33) ∗  1 1∗  1 ∗ s–2 where f ∈ H ,which,due to,e.g., [31, Theorem 2.10], can be in turn represented as 1∗ ∗ s– f =–γ  with some  ∈ H (∂). Then (6.30)issatisfied, and 1∗ ∗ ∗ + –1 ˜ ˜ ˇ F = T (f , aF )= γ f – (aF ) 2 ∗ 1 ∗ 1 ∗ ∗ –1 –1 ∗ = γ f =– γ γ  =– , (6.34) 1∗ ∗ ∗ –1 ∗ ∗ ∗ –1 –s because (γ ) γ  , w = γ  , γ w =  , w for any w ∈ H (∂). Hence ∗ ∂ ∗  ∗ ∂ (6.33)reduces to (6.26). Now (6.28)can be writteninthe form W (a )= F in , (6.35) where ˜ ˇ F := –aF + P f =–aF + P (aF )+ γ F (6.36) ∗ 1 1 2 is a harmonic function in  due to (6.29). The trace of equation (6.35)gives – a + W (a )= γ F on ∂. (6.37) 1 1 1 s– s– 2 2 Since the operator – I + W : H (∂) → H (∂) is an isomorphism (see Theo- rem 7.3), this implies –1 1 1 = – I + W γ F a 2 –1 1 1 + ∗ = – I + W γ –aF + P (aF )+ γ F , 1 2 a 2 which coincides with (6.27). Relations (6.26)and (6.27)can be writtenas (f ,  )= C (F , F ), where C : H () × ∗ ∗ ∗ 1 2 ∗ 3 1 s– s–2 s– 2 2 H (∂) → H () ×H (∂) is a linear continuous operator, as claimed. We still have to check that the functions f and  ,given by (6.26)and (6.27), satisfy equations (6.24) ∗ ∗ and (6.25). Indeed,  given by (6.27) satisfies equation (6.37)with F given by (6.36), and + + thus γ W (a )= γ F .Since both W (a )and F are harmonic functions belonging Mikhailov Boundary Value Problems (2018) 2018:87 Page 38 of 52 to the space H (), this implies (6.35)and,by (6.26), also (6.24). Finally, (6.26)implies by (6.34)that(6.32) is satisfied, and adding (6.31)toitleads to (6.25). s–2 Let us now prove that the operator C is unique. Indeed, let a couple (f ,  ) ∈ H () × ∗ ∗ ∗ s– H (∂) be a solution of linear system (6.24)–(6.25)with F =0 and F =0. Then (6.30) 1 2 s–2 s–2 + ˜ ˜ ˜ implies that r f =0 in , i.e., f ∈ H ⊂ H (). Hence, (6.32)reduces to 0 = T (f ,0) ∗ ∗ ∗ on ∂. By the first Green identity (2.15)thisgives + + 2–s ˜ ˜ 0= T (f ,0), γ v = f , v ∀ v ∈ H (), ∗ ∗ which implies f =0 in R . Finally, (6.27)gives  = 0. Hence, any solution of non- ∗ ∗ homogeneous linear system (6.24)–(6.25) has only one solution, which implies the unique- ness of the operator C . 1 3 Theorem 6.7 Let  be a bounded simply connected Lipschitz domain, < s < , and a ∈ 2 2 C () with σ = max{1, s}. The co-kernel of operator (6.14) is spanned over the functional ∗1 g := (0, 1) (6.38) 3 3 s s– ∗ –s –s ∗1 + 0 2 2 in [H () × H (∂)] = H () × H (∂), i.e., g (F , F )= F , γ u , where 1 2 2 ∂ u =1. Proof Let us consider the equation N U =(F , F ) , i.e., the BDIE system (N1 )for 1 2 (u, ϕ) ∈ H () × H (∂), u + Ru + W ϕ = F in , (6.39) + ∇ T A u; aRu + L (aϕ)= F on ∂, (6.40) s s– with arbitrary (F , F ) ∈ H () × H (∂). By Lemma 6.6 the right-hand side of the sys- 1 2 tem can be presented in the form (6.24)–(6.25), i.e., system (6.39)–(6.40)reduces to u + Ru + W(ϕ +  )= Pf in , (6.41) ∗ ∗ + ∇ + ˜ ˜ T A u; aRu + L (aϕ + a )= T (f ; Pf )on ∂, (6.42) ∗ ∗ ∗ s–2 s– where the couple (f ,  ) ∈ H () × H (∂)is given by (6.26)–(6.27). Up to the no- ∗ ∗ tations, system (6.41)–(6.42)isthe same as (6.4)–(6.5) with the right-hand side given by (6.6), where ψ =0. First, let s = 1. Then Theorems 6.1 and 6.3 imply that BDIE system (6.41)–(6.42)and hence (6.39)–(6.40) are solvable if and only if 0 ∗ 0 ˜ ˇ f , u = (aF )+ γ F , u ∗  1 2 ∗ 0 = ∇· E ∇(aF )+ γ F , u 1 2 n 0 + 0 =– E ∇(aF ), ∇u + F , γ u 1 2 R ∂ + 0 = F , γ u = 0, (6.43) ∂ Mikhailov Boundary Value Problems (2018) 2018:87 Page 39 of 52 0 n ∗1 where we took into account that u =1 in R . Thus the functional g defined by (6.38) generates the necessary and sufficient solvability condition of equation N U =(F , F ) . 1 2 ∗1 Hence g is a basis of the co-kernel of N (and thus the kernel of the operator N adjoint to N )for s =1. 1 3 Let us now choose any s ∈ ( , ). By Theorem 6.4,operator(6.14) and thus its adjoint 2 2 are Fredholm with zero index. We already proved that, at s = 1, the kernel of the adjoint ∗1 operator is spanned over g . For any fixed coefficient a ∈ C (), the operator 1   3 s s – s s – 2 2 N : H () × H (∂) → H () × H (∂) (6.44) is continuous for any s ∈ ( , σ]and particularly for s = s and s = 1. Then Lemma 7.5 implies that the co-kernel of operator (6.44)isthe same for s = s and s = 1 and is spanned ∗1 over g . 1 3 Lemma 6.8 Let  be a bounded simply connected Lipschitz domain, < s < , and 2 2 2 s s– a ∈ C (). For any couple (F , F ) ∈ H () × H (∂), there exists a unique couple + 1 2 s– s–2 (f ,  ) ∈ H () × H (∂) such that ∗∗ ∗∗ F = Pf – W  in , (6.45) 1 ∗∗ ∗∗ + + ˜ ˜ ˜ F = T (f + E R f ; Pf )– T W  on ∂. (6.46) 2 ∗∗  ∗ ∗∗ ∗∗ ∗∗ 3 1 s s– s–2 s– 2 2 Moreover,(f ,  )= C (F , F ), and C : H () × H (∂) → H () × H (∂) is ∗∗ ∗∗ ∗∗ 1 2 ∗∗ a linear continuous operator given by ∗ + f = (aF )+ γ F + γ F ∂ a , (6.47) ∗∗  1 2 1 n –1 1 1 + ∗ + = – I + W γ –aF + P (aF )+ γ F + γ F ∂ a . (6.48) ∗∗ 1 2 1 n a 2 s–2 s– Proof Let us first assume that there exist (f ,  ) ∈ H () × H (∂)satisfying equa- ∗∗ ∗∗ tions (6.45)and (6.46) and prove that they are then expressed in terms of F and F by 1 2 (6.47)–(6.48). Let us rewrite (6.45)as F – Pf =–W  in , (6.49) 1 ∗∗ ∗∗ Multiplying (6.49)by a and applying the Laplace operator to it, we obtain ˜ ˜ (aF – P f )= (aF )– f =– W (a )=0 in , (6.50) ∗∗ 1 ∗∗ ∗∗ which means that (aF )= r f in  (6.51) 1  ∗∗ s,0 and aF – P f ∈ H (; ). By equality (6.49) and the continuity of operator (3.73)in ∗∗ 1,0 Theorem 3.5,wealsohave F – Pf ∈ H (; A), which implies that the canonical co- 1 ∗∗ Mikhailov Boundary Value Problems (2018) 2018:87 Page 40 of 52 normal derivative T (F – Pf ) is well defined. Applying the canonical co-normal deriva- 1 ∗∗ tive operator T to both sides of equation (6.49), we obtain + + + ˜ ˜ ˜ ˜ –T W  = T (F – Pf )= T A(F – Pf ); F – Pf ∗∗ 1 ∗∗ 1 ∗∗ 1 ∗∗ ˚ ˜ ˜ = T E ∇· a∇(F – Pf ) ; F – Pf 1 ∗∗ 1 ∗∗ ˚ ˜ ˚ ˜ ˜ = T E (aF – P f )– E ∇· (F – Pf )∇a ; F – Pf ∗∗  1 ∗∗ 1 ∗∗ ˚ ˚ ˜ ˜ = T –E ∇· (F ∇a)– E R f ; F – Pf , (6.52) 1  ∗ ∗∗ 1 ∗∗ where (6.50) and the third relation in (3.16) were taken into account. Substituting this into (6.46), we obtain ˜ ˚ F = T f – E ∇· (F ∇a), F on ∂. (6.53) 2 ∗∗  1 1 Due to (6.51), we can represent ˜ ˇ ˜ ˚ f = (aF )+ f = ∇· E ∇(aF )– γ  , (6.54) ∗∗  1 1∗  1 ∗∗ s–2 where f ∈ H ,which,due to,e.g., [31, Theorem 2.10], can be in turn represented as 1∗ ∗ s– f =–γ  with some  ∈ H (∂). Then (6.51)issatisfied, and 1∗ ∗∗ ∗∗ + –1 ˜ ˚ ˜ ˚ ˇ F = T f – E ∇· (F ∇a), F = γ f – E ∇· (F ∇a)– AF 2 ∗∗  1 1 ∗∗  1 1 –1 ∗ ˚ ˚ ˚ = γ ∇· E ∇(aF )– γ  – E ∇· (F ∇a)– ∇· E (a∇F ) 1 ∗∗  1  1 –1 ∗ + ˚ ˚ = γ ∇· E (F ∇a)– γ  – E ∇· (F ∇a) =– – γ F ∂ a, (6.55) 1 ∗∗  1 ∗∗ 1 n –s because for any w ∈ H (∂), –1 ∗ ˚ ˚ γ ∇· E (F ∇a)– γ  – E ∇· (F ∇a) , w 1 ∗∗  1 ∗ –1 ˚ ˚ = ∇· E (F ∇a)– γ  – E ∇· (F ∇a), γ w 1 ∗∗  1 –1 ∗ –1 –1 ˚ ˚ = ∇· E (F ∇a), γ w – γ  , γ w – E ∇· (F ∇a), γ w 1 n ∗∗  1 –1 –1 =– E (F ∇a), ∇γ w –  , w + F ∇a, ∇γ w 1 n ∗∗ ∂ 1 + + –1 + – n · γ (F ∇a), γ γ w =– γ F ∂ a, w –  , w . 1 1 n ∗∗ ∂ ∂ ∂ Hence (6.53)reduces to  =–F –(γ F )∂ a,and (6.54)to(6.47). ∗∗ 2 1 n Now (6.49)can be writteninthe form W (a )= F in , (6.56) ∗∗ where ∗ + ˜ ˇ F := –aF + P f =–aF + P (aF )+ γ F + γ F ∂ a (6.57) ∗∗ 1 1 2 1 n is a harmonic function in  due to (6.50). The trace of equation (6.56)gives – a + W (a )= γ F on ∂. (6.58) ∗∗ ∗∗ 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 41 of 52 1 1 1 s– s– 2 2 Since the operator – I + W : H (∂) → H (∂) is an isomorphism (see Theo- rem 7.3), this implies –1 1 1 = – I + W γ F ∗∗ a 2 –1 1 1 + ∗ + = – I + W γ –aF + P (aF )+ γ F + γ F ∂ a , 1 2 1 n a 2 which coincides with (6.48). Relations (6.47)and (6.48)can be writtenas (f ,  )= C (F , F ), where C : H () × ∗∗ ∗∗ ∗∗ 1 2 ∗∗ 3 1 s– s–2 s– 2  2 H (∂) → H () × H (∂) is a linear continuous operator, as claimed. We still have to check that the functions f and  given by (6.47)and (6.48)satisfy equa- ∗∗ ∗∗ tions (6.45)and (6.46). Indeed,  given by (6.48) satisfies equation (6.58), and thus ∗∗ + + γ W (a )= γ F .Since both W (a )and F are harmonic functions belonging ∗∗ ∗∗ to the space H (), this implies (6.56)–(6.57)and by (6.47)also(6.45). Finally, (6.47)im- plies by (6.55)that(6.53) is satisfied, and adding (6.52)toitleads to (6.46). Let us now prove that the operator C is unique. Indeed, let a couple (f ,  ) ∈ ∗∗ ∗∗ ∗∗ s– s–2 H () × H (∂) be a solution of linear system (6.45)–(6.46)with F =0 and F =0. 1 2 s–2 s–2 ˜ ˜ Then (6.51)implies that r f =0 in , i.e., f ∈ H ⊂ H (). Hence, (6.53)reduces to ∗∗ ∗∗ 0= T (f ,0) on ∂. By the first Green identity (2.15)thisgives ∗∗ + + 2–s ˜ ˜ 0= T (f ,0), γ v = f , v ∀ v ∈ H (), ∗∗ ∗∗ which implies f =0 in R . Finally, (6.48)gives  = 0. Hence, non-homogeneous linear ∗∗ ∗∗ system (6.45)–(6.46) has only one solution, which implies the uniqueness of the opera- tor C . ∗∗ 1 3 Theorem 6.9 Let  be a bounded simply connected Lipschitz domain, < s < , and a ∈ 2 2 C (). The co-kernel of operator (6.15) is spanned over the functional ∗1 + g := γ ∂ a,1 (6.59) 3 3 s s– ∗ –s –s 2 2 in [H () × H (∂)] = H () × H (∂), i.e., ∗1 + + 0 g (F , F )= γ F ∂ a + F , γ u , 1 2 1 n 2 where u =1. Proof Let us consider the equation N U =(F , F ) , i.e., the BDIE system (N1) for (u, ϕ) ∈ 1 2 s s– H () × H (∂), u + Ru + W ϕ = F in , (6.60) + + + T Ru + T W ϕ = F on ∂, (6.61) 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 42 of 52 s s– with arbitrary (F , F ) ∈ H () × H (∂). By Lemma 6.8 the right-hand side of the sys- 1 2 tem has form (6.45)–(6.46), i.e., system (6.60)–(6.61)reduces to u + Ru + W(ϕ +  )= Pf in , (6.62) ∗∗ ∗∗ + + + ˜ ˚ ˜ ˜ T Ru + T W(ϕ +  )= T (f + E R f , Pf )on ∂, (6.63) ∗∗ ∗∗  ∗ ∗∗ ∗∗ s–2 s– where the couple (f ,  ) ∈ H () × H (∂)is given by (6.47)–(6.48). Up to the no- ∗∗ ∗∗ tations, system (6.62)–(6.63)isthe same as (6.7)–(6.8) with the right-hand side given by (6.9), where ψ =0. First, let s = 1. Then Theorems 6.1 and 6.3 imply that BDIE system (6.62)–(6.63)and hence (6.60)–(6.61) are solvable if and only if 0 ∗ + 0 f , u = (aF )+ γ F + γ F ∂ a , u ∗∗  1 2 1 n ∗ + 0 = ∇· E ∇(aF )+ γ F + γ F ∂ a , u 1 2 1 n 0 + + 0 =– E ∇(aF ), ∇u + F + γ F ∂ a, γ u 1 n 2 1 n R ∂ + + 0 = γ F ∂ a + F , γ u =0, 1 n 2 0 n ∗1 wherewetookintoaccountthat u =1 in R . Thus the functional g defined by (6.59) generates the necessary and sufficient solvability condition of equation N U =(F , F ) . 1 2 ∗1 1 Hence g is a basis of the co-kernel of N (and thus the kernel of the operator adjoint to N )for s =1. 1 3 Let now s ∈ ( , ). By Theorem 6.4 operator (6.15) and thus its adjoint are Fredholm with 2 2 zero index. We already proved that, at s = 1, the kernel of the adjoint operator is spanned ∗1 1 3 over g . Then Lemma 7.5 implies that the kernel is the same for any s ∈ ( , ). 2 2 To find the co-kernel of operator (6.16), we need some auxiliary assertions. Lemma 6.10 and Theorem 6.11 were proved in [33, Lemma 6.4 and Theorem 6.5] for the infinitely smooth coefficient a and boundary ∂. We further only slightly modify these proofs for the non-smooth coefficients and Lipschitz boundary. Lemma 6.10 Let  be a bounded simply connected Lipschitz domain, s > , a ∈ C (), s–2 and f ∈ H (). If r Pf =0 in , (6.64) then f =0 in R . Proof Multiplying (6.64)by a, taking into account (3.16), and applying the Laplace oper- s–2 3 ˜ ˜ ˜ ator, we obtain r f =0, which means f ∈ H .If s ≥ ,then f = 0 by Theorem 2.10 from 1 3 s– ∗ [31]. If < s < , then by the same theorem there exists v ∈ H (∂)such that f = γ v. 2 2 ∗ n This gives Pf = Pγ v =–Vv in R ;see (3.53). Then (6.64)reduces to Vv =0 in ,which implies v =0 on ∂ by Lemma 4.8(i), and thus f =0 in R .  Mikhailov Boundary Value Problems (2018) 2018:87 Page 43 of 52 1 3 Theorem 6.11 Let  be a bounded simply connected Lipschitz domain, < s < , and 2 2 a ∈ C (). The operator s–2 s r P : H () → H () (6.65) and its inverse –1 s s–2 (r P) : H () → H () (6.66) are continuous, and –1 –1 + ∗ –1 + n s (r P) g = E I – r V V γ – γ V γ (ag) in R , ∀ g ∈ H (). (6.67) Proof The continuity of (6.65)isgiven by Theorem 3.2. By Lemma 6.10 operator (6.65)is injective. Let us prove its surjectivity. To this end, for arbitrary g ∈ H (), let us consider s–2 the following equation with respect to f ∈ H (): r P f = g in . (6.68) Let g ∈ H () be the (unique) solution of the following Dirichlet problem: g =0 in , 1 1 + + –1 + γ g = γ g, which can be particularly presented as g = V V γ g; see, e.g., [11]orproof 1 1 s + of Lemma 2.6 in [31]. Let g := g – r g .Then g ∈ H ()and γ g =0, and thus g can be 0  1 0 0 0 uniquely extended to E g ∈ H (). Thus by (3.53)equation(6.68)takes form ∗ –1 + r P f + γ V γ g = g in . (6.69) s–2 n Any solution f ∈ H () of the corresponding equation in R , ∗ –1 + n ˜ ˚ P f + γ V γ g = E g in R , (6.70) evidently solves (6.69). If f solves (6.70), then applying the Laplace operator to (6.70), we obtain ∗ –1 + –1 + ∗ –1 + n ˜ ˜ ˚ ˚ f = Qg := E g – γ V γ g = E g – r V V γ g – γ V γ g in R . (6.71) On the other hand, substituting f given by (6.71)into(6.70) and taking into account that ˜ ˜ ˜  ˜ h = h for any h ∈ H (), s ∈ R,weobtainthat Qg is indeed a solution of equation (6.70)and thus of (6.69). By Lemma 6.10 the solution of (6.69) is unique, which means –1 ˜ ˜ that the operator Q is inverse to operator (6.65), i.e., Q =(r P) .Since is a continuous s s–2 –1 operator from H ()to H (), equation (6.71) implies that the operator (r P) = Q : s s–2 1 H () → H () is continuous. The relations P = P and a(x) ≥ a >0 then imply the min invertibility of operator (6.65) and ansatz (6.67). 1 3 Theorem 6.12 Let  be a bounded simply connected Lipschitz domain, < s < , and a ∈ 2 2 C () with σ = max{1, s}. The co-kernel of operator (6.16) is spanned over the functional +∗  –1 + 0 –aγ ( + W )V γ u ∗2 g := (6.72) –1 + 0 –a( – W )V γ u 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 44 of 52 1 1 s s– ∗ –s –s 2  2 in [H () × H (∂)] = H () × H (∂), i.e., ∗2 g (F , F ) 1 2 1 1 +∗  –1 + 0  –1 + 0 = –aγ + W V γ u , F + –a – W V γ u , F , 1 2 2 2 where u (x)=1. Proof Let us consider the equation N U =(F , F ) , i.e., the BDIE system (N2), 1 2 u + Ru + W ϕ = F in , (6.73) ϕ + γ Ru + W ϕ = F on ∂, (6.74) 1 1 s s– s s– 2 2 with arbitrary (F , F ) ∈ H () × H (∂)for (u, ϕ) ∈ H () × H (∂). 1 2 Introducing the new variable ϕ = ϕ –(F – γ F ), BDIE system (6.73)–(6.74)takes form 2 1 u + Ru + W ϕ = F in , (6.75) +  + ϕ + γ Ru + W ϕ = γ F on ∂, (6.76) where + s F = F – W F – γ F ∈ H (). 1 2 1 On the other hand, by Theorem 6.11, we can always represent F = Pf ,with –1 + +∗ –1 +  s–2 f = E I – r V V γ – γ V γ aF ∈ H (). For F = Pf , the right-hand side of BDIE system (6.73)–(6.74)isthe same as in (6.12)with ˜ ˜ f = f and ψ =0. ∗ 0 First, let s = 1. Then Theorems 6.1 and 6.3 imply that BDIE system (6.75)–(6.76)issolv- able if and only if 0 –1 + +∗ –1 +  0 ˜ ˚ f , u = E I – r V V γ – γ V γ aF , u 1 n –1 +  0 –1 +  + 0 = E I – r V V γ aF , u – V γ aF , γ u 1 n R ∂ +  –1 + 0 =– γ aF , V γ u + + –1 + 0 =– γ (aF )+(aF ) – W a F – γ F , V γ u 1 2 2 1 1 1 +∗  –1 + 0  –1 + 0 =– F , aγ + W V γ u – F , a – W V γ u 1 2 2 2 = 0. (6.77) ∗2 Thus the functional g defined by (6.72) generates a necessary and sufficient solvability 2  ∗2 condition of equation N U =(F , F ) .Hence g is a basis of the co-kernel of operator 1 2 (6.16)for s =1. Mikhailov Boundary Value Problems (2018) 2018:87 Page 45 of 52 1 3 Let us now choose any s ∈ ( , ). By Theorem 6.4 operator (6.16) and thus its adjoint 2 2 are Fredholm with zero index. We already proved that, at s = 1, the kernel of the adjoint ∗2 σ operator is spanned over g . For any fixed coefficient a ∈ C (), the operator 1   1 2 s s – s s – 2 2 N : H () × H (∂) → H () × H (∂) (6.78) is continuous for any s ∈ ( , σ]and particularly for s = s and s = 1. Then Lemma 7.5 implies that the co-kernel of operator (6.78)isthe same for s = s and s = 1 and is spanned ∗2 over g . Theorems 6.3, 6.5,and 6.7 (or 6.9) imply the following extension of Theorem 6.1 to the 1 3 range < s < . 2 2 1 3 Corollary 6.13 Let  be a bounded simply connected Lipschitz domain, < s < , f ∈ 2 2 s– s–2 σ H (), ψ ∈ H (∂), and a ∈ C () with σ = max{1, s}. The homogeneous Neumann problem (6.1)–(6.2) admits only one linearly independent 0 s solution u =1 in H (). The non-homogeneous Neumann problem (6.1)–(6.2) is solvable in H () if and only if condition (6.3) is satisfied. Proof Assuming that a function u is a solution of the homogeneous Neumann problem, by Theorem 6.3 the couple (u, ϕ)=(u, γ ϕ) solves the homogeneous BDIE system (N1 ), and then Theorem 6.7 implies that u is spanned over u =1. Assume that solvability condition (6.3) is satisfied. Then the right-hand side (6.6)ofthe ∗1 + 0 BDIE system (N1 ) satisfies its solvability condition g (F , F )= F , γ u =0 given 1 2 2 ∂ by Theorem 6.7. Indeed, due to the first Green identities (2.15)and (2.18) applied to the operator and Remark 2.7,since V ψ is a harmonic function in  and u =1, we obtain + 0 +  + 0 ˜ ˜ F , γ u = T (f ; P f )– ψ + W ψ , γ u 0 0 + + + 0 ˜ ˜ = T (f ; P f )– ψ + T V ψ , γ u 0 0 + 0 ˜ ˇ ˜ = f , u + E P f , u – ψ , γ u 0 0 ˜ ˇ V ψ , u + E V ψ , u 0 + 0 = f , u – ψ , γ u . (6.79) Hence the BDIE system (N1 ) is solvable, implying solvability of the Neumann BVP due to Theorem 6.3(ii). This proves that condition (6.3)issufficient. Let us now assume that there exists a solution of the Neumann BVP. Hence Theo- rem 6.3(i) implies that the BDIE system (N1 ) with the right-hand side (6.9)issolvable, + 0 implying that its solvability condition F , γ u =0 is satisfied. Then (6.79)implies 2 ∂ condition (6.3), proving that it is necessary. 6.3 Perturbed (stabilised) segregated BDIE systems for the Neumann problem Theorem 6.5 implies that even when the solvability condition (6.3)issatisfied, the solutions of BDIE systems (N1 ), (N1), and (N2) are not unique, and moreover, the 1 2 BDIE left-hand side operators N , N ,and N , have non-zero kernels and thus are Mikhailov Boundary Value Problems (2018) 2018:87 Page 46 of 52 not invertible. To find a solution (u, ϕ) from uniquely solvable BDIE systems with continuously invertible left-hand side operators, let us consider, following [28], some stabilised BDIE systems obtained from (N1 ), (N1), and (N2) by finite-dimensional operator perturbations. Note that other choices of the perturbing operators are also pos- sible. N  0 We further use the notations U =(u, ϕ) , U = (1, 1) ,and |∂| := dS. Let us introduce the perturbed counterparts of the BDIE systems (N1 ), (N1), and (N2): N N1 1 N N1 2 N N2 ˆ ˆ ˆ N U = F , N U = F , N U = F , (6.80) 1 1 1 2 2 2 ˆ ˚ ˆ ˚ ˆ ˚ where N := N + N , N := N + N , N := N + N ,and 1 0 N 1 N 0 N 1 ˚ ˚ N U (y)= N U (y):= g U G (y)= ϕ(x) dS , (6.81) |∂| 1 that is, 1 0 0 N 1 g U := ϕ(x) dS, G (y):= , (6.82) |∂| 1 whereas –1 1 a (y) 2 N 0 N 2 N U := g U G = ϕ(x) dS , + –1 |∂| γ a (y) 0 N that is, g (U ) isasin(6.82), and –1 a (y) G (y):= . + –1 γ a (y) 1 3 Theorem 6.14 Let  be a bounded simply connected Lipschitz domain, < s < , and 2 2 σ = max{1, s}. (i) The following operators are continuous and continuously invertible: 1 3 s s– s s– σ 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (), (6.83) 1 3 1 s s– s s– 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (), (6.84) 1 1 2 s s– s s– σ 2 2 N : H () × H (∂) → H () × H (∂) if a ∈ C (). (6.85) ∗1 N1 ∗1 N1 ∗2 N2 (ii) If the conditions g (F )=0, g (F )=0, or g (F )=0 are satisfied, then the unique solutions of the perturbed BDIE systems in (6.80) give the solutions U of the corresponding original BDIE systems (N1 ), (N1), and (N2) such that 1 1 0 N + g U = ϕ dS = γ udS =0. |∂| |∂| ∂ ∂ Mikhailov Boundary Value Problems (2018) 2018:87 Page 47 of 52 ∗1 ∗1 Proof For the functional g given by (6.38)inTheorem 6.7, g (G )= |∂|. Similarly, ∗1 ∗1 1 for the functional g given by (6.59)inTheorem 6.9, g (G )= |∂|. ∗2 –1 For the functional g given by (6.72)inTheorem 6.12, since the operator V : 1 1 2 2 H (∂) → H (∂) is positive definite and u (x) = 1, there exists a positive constant C such that ∗2 2 +∗  –1 + 0 –1 0 g G = –aγ + W V γ u , a u –1 + 0 + –1 0 + –a – W V γ u , γ a u 1 1 –1 + 0  –1 + 0 + 0 =– + W V γ u + – W V γ u , γ u 2 2 –1 + 0 + 0 =– V γ u , γ u 2 2 + 0 + 0 ≤ –C γ u 1 ≤ –C γ u L (∂) 2 2 H (∂) =–C|∂| < 0. (6.86) 0 0 On the other hand, g (U )=1. Hence Theorem 7.4 from [28]implies theclaimsofthe theorem. 7 Auxiliary assertions We provide here some auxiliary results used in the main text. 1 3 s σ Theorem 7.1 Let < s < , u ∈ H (), a ∈ C () with σ = max{1, s}, Au = r finaninte- 2 2 s–2 2 rior or exterior Lipschitz domain  for some f ∈ H (). Let {f }∈ H () be a sequence ˜ ˚ such that f – E f → 0 as k →∞. s–2 H () s,– Then there exists a sequence {u }∈ H (; A) such that Au = f in  and u – k k k + + u → 0 as k →∞. Moreover, T (u )– T (f ; u) 3 → 0 as k →∞. k H () k s– H (∂) Proof Let us consider the Dirichlet problem Au = f in , (7.1) k k + + γ u = γ u on ∂, (7.2) s D –1 By Corollary 5.5theuniquesolutionofproblem(7.1)–(7.2)in H ()is u =(A ) (f , ϕ ) , k k k D –1 s–2 s– s where (A ) : H () × H (∂) → H () is a continuous operator. Hence the func- s 2 tions u converge to u in H ()as k →∞.Since Au = f ∈ H (), we obtain that k k k • s,– + u ∈ H (; A) and the canonical conormal derivative T u is well defined. Then sub- k k tracting (2.16)for u from (2.12), we obtain + + –1 ˜ ˜ ˇ T (f ; u)– T u = γ f – E f + A (u – u ) . k  k  k Hence + + ˜ ˜ ˚ T (f ; u)– T u 3 ≤ C f – E f s–2 + C u – u (7.3) k  k  1 k H () s– H () H (∂) Mikhailov Boundary Value Problems (2018) 2018:87 Page 48 of 52 for some positive C and C . Since the right-hand side of (7.3)tends to zero as k →∞,so does the left-hand side. 1 1 – – 2 s–2 2 Note that since D() ⊂ H ()is dense in H (), the sequence {f }∈ H ()from • • the hypotheses of Theorem 7.1 does always exist. The following multiplication theorem is well known; see, e.g., [15, Theorems 1.4.1.1, 1.4.1.2], [54, Theorem 2(b)], [1, Theorems 1.9.1, 1.9.2, 1.9.5], [32, Theorem 3.2]. Theorem 7.2 Let  be an open set. (i) If g ∈ L ( ), then gv ∈ L ( ) and gv ≤ c g v for every ∞ 0 2 0 L () L ( ) L ( ) 2 0 2 0 v ∈ L ( ). 2 0 |σ |–1,1 σ (ii) If σ is a non-zero integer and g ∈ C ( ), then gv ∈ H ( ) for every 0 0 v ∈ H ( ), and gv σ ≤ c g v σ . 0 H () |σ |–1,1 H ( ) C ( ) 0 (iii) If σ is a non-integer, |σ | = m + θ, where m is a non-negative integer and 0< θ <1, m,η σ then for g ∈ C ( ) with θ < η <1, we have gv ∈ H ( ) and 0 0 gv σ ≤ c g v σ for every v ∈ H ( ). m,η H () H ( ) 0 C ( ) 0 In all cases, c is a positive constant independent of g, v, or  . Theorem 7.3 Let  be a bounded simply connected Lipschitz domain, and let 0 ≤ σ ≤ 1. The operators σ –1 σ V : H (∂) → H (∂), (7.4) σ σ – I + W : H (∂) → H (∂), (7.5) –σ –σ – I + W : H (∂) → H (∂) (7.6) are isomorphisms, and the operators σ σ I + W : H (∂) → H (∂), (7.7) –σ –σ I + W : H (∂) → H (∂), (7.8) σ σ –1 L : H (∂) → H (∂) (7.9) are Fredholm with zero index. Proof The properties of the boundary integral operators (7.4)–(7.9) related to the har- monic layer potential are well known; see, e.g., [51], [39, Theorem 4.1], [14, Theorem 8.1] for the invertibility of operators (7.4)–(7.6) and the Fredholm properties of operators (7.7)–(7.8). The Fredholm property of operator (7.9)for σ = is also well known; see, e.g., [27, Theorem 7.8]. Then the corresponding result for 0 ≤ σ ≤ 1 can be proved as in [27, Theorem 7.17] by using a sharper regularity result from [11, Theorem 3]. Theorem 7.4 further is implied by [28, Lemma 2] (see also [50,Sect. 21],[49, Sect. 21.4], ∗ ∗ where the particular case h (x )= x ˚ (h )= δ has been considered). Another approach, j j ij i i although with hypotheses similar to those in Theorem 7.4,ispresented in [17, Lemma 4.8.24]. Mikhailov Boundary Value Problems (2018) 2018:87 Page 49 of 52 Theorem 7.4 Let B and B be two Banach spaces. Let A : B → B be a linear Fredholm 1 2 1 2 ∗ ∗ ∗ operator with zero index, and let A : B → B be the adjoint operator with dim ker A = 2 1 ∗ n ∗ ∗ n ∗ dim ker A = n < ∞, where ker A = span{x ˚ } ⊂ B and ker A = span{x ˚ } ⊂ B . Let i 1 i=1 i i=1 2 A x := h h (x), 1 i i=1 ∗ ∗ where h and h (i = 1,..., n) are elements from B and B , respectively, such that i 2 i 1 ∗ ∗ det h (x ˚ ) =0, det x ˚ (h ) =0, i, j = 1,..., n. (7.10) j j i i Then: (i) the operator A – A : B → B is an isomorphism; 1 2 (ii) if y ∈ B satisfies the solvability conditions x ˚ (y)=0, i = 1,..., n, (7.11) of the equation Ax = y, (7.12) then the unique solution x of equation (A – A )x = y, (7.13) is a solution of equation (7.12) such that h (x)=0 (i = 1,..., k). (7.14) (iii) Vice versa, if x is a solution of equation (7.13) satisfying conditions (7.14), then conditions (7.11) are satisfied for the right-hand side y of equation (7.13), and x is a solution of equation (7.12) with the same right-hand side y. Note that more results about finite-dimensional operator perturbations are available in [28]. The following known result (see, e.g., [42, Lemma 11.9.21]) is useful for us. Lemma 7.5 Let X , X and Y , Y , be Banach spaces such that the embeddings X → X 1 2 1 2 1 2 and Y → Y are continuous, and the embedding Y → Y has a dense range. Assume that 1 2 1 2 T : X → Y and T : X → Y are Fredholm operators with the same index, ind(T : X → 1 1 2 2 1 Y )= ind(T : X → Y ). Then Ker{T : X → Y } = Ker{T : X → Y }. 1 2 2 1 1 2 2 8 Concluding remarks The Dirichlet and Neumann problems on a bounded Lipschitz domain for a variable- s–2 coefficient second-order PDE with general right-hand side functions from H ()and 1 3 s–2 H (), < s < , respectively, were equivalently reduced to three direct segregated 2 2 Mikhailov Boundary Value Problems (2018) 2018:87 Page 50 of 52 boundary-domain integral equation systems for each of the BVPs. This involved system- atic use of the generalised co-normal derivatives. The operators associated with the left- hand sides of all the BDIE systems were analysed in the corresponding Sobolev spaces. It was shown that the operators of the BDIE systems for the Dirichlet problem are con- tinuous and continuously invertible. For the Neumann problem, the BDIE system opera- tors are continuous but only Fredholm with zero index; their kernels and co-kernels were analysed, and appropriate finite-dimensional perturbations were constructed to make the perturbed (stable) operators invertible and provide a solution of the original BDIE systems and the Neumann problem. The same approach can be implemented to extend to the general PDE right-hand sides, non-smooth coefficients and Lipschitz domains: the BDIE systems for the mixed prob- lems, unbounded domains, BDIEs of more general scalar PDEs and the systems of PDEs, and the united and localised BDIEs, for which the analysis is now available for the right- hand sides only from L (), with smooth coefficients and smooth domain boundaries; see [2–10, 13, 30, 36, 37]. Acknowledgements Not applicable. Funding This research was supported by the grants EP/H020497/1: “Mathematical Analysis of Localized Boundary-Domain Integral Equations for Variable-Coefficient Boundary Value Problems” and EP/M013545/1: “Mathematical Analysis of Boundary-Domain Integral Equations for Nonlinear PDEs” from the EPSRC, UK. Abbreviations BDIE, Boundary-domain integral equation; BVP, Boundary value problem; PDE, Partial differential equation. Availability of data and materials Data sharing not applicable to this article as no datasets were generated or analysed during the current study. Competing interests The author declares that he has no competing interests. Authors’ contributions The paper was written by the author personally. The author read and approved the final manuscript. Publisher’s Note Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. Received: 12 January 2018 Accepted: 30 April 2018 References 1. Agranovich, M.S.: Sobolev Spaces, Their Generalizations, and Elliptic Problems in Smooth and Lipschitz Domains. Springer, Cham (2015) 2. Ayele, T.G., Mikhailov, S.E.: Analysis of two-operator boundary-domain integral equations for variable-coefficient mixed BVP. Eurasian Math. J. 2(3), 20–41 (2011) 3. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, I: Equivalence and invertibility. J. Integral Equ. Appl. 21(4), 499–543 (2009) 4. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of some localized boundary-domain integral equations. J. Integral Equ. Appl. 21(3), 405–445 (2009) 5. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct boundary-domain integral equations for a mixed BVP with variable coefficient, II: Solution regularity and asymptotics. J. Integral Equ. Appl. 22(1), 19–37 (2010) 6. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of segregated boundary-domain integral equations for variable-coefficient problems with cracks. Numer. Methods Partial Differ. Equ. 27(1), 121–140 (2011). https://doi.org/10.1002/num.20639 7. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Localized direct segregated boundary-domain integral equations for variable-coefficient transmission problems with interface crack. Mem. Differ. Equ. Math. Phys. 52, 17–64 (2011) 8. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Analysis of direct segregated boundary-domain integral equations for variable-coefficient mixed BVPs in exterior domains. Anal. Appl. 11(4), 1350006 (2013). https://doi.org/10.1142/S0219530513500061 Mikhailov Boundary Value Problems (2018) 2018:87 Page 51 of 52 9. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Localized boundary-domain singular integral equations based on harmonic parametrix for divergence-form elliptic PDEs with variable matrix coefficients. Integral Equ. Oper. Theory 76(4), 509–547 (2013). https://doi.org/10.1007/s00020-013-2054-4 10. Chkadua, O., Mikhailov, S.E., Natroshvili, D.: Localized boundary-domain singular integral equations of Dirichlet problem for self-adjoint second order strongly elliptic PDE systems. Math. Methods Appl. Sci. 40, 1817–1837 (2017). https://doi.org/10.1002/mma.4100 11. Costabel, M.: Boundary integral operators on Lipschitz domains: elementary results. SIAM J. Math. Anal. 19, 613–626 (1988) 12. Dautray, R., Lions, J.L.: Mathematical Analysis and Numerical Methods for Science and Technology Vol. 4: Integral Equations and Numerical Methods. Springer, Berlin (1990) 13. Dufera, T.T., Mikhailov, S.E.: Analysis of boundary-domain integral equations for variable-coefficient Dirichlet BVP in 2D. In: Constanda, C., Kirsh, A. (eds.) Integral Methods in Science and Engineering: Theoretical and Computational Advances. Springer, Boston (2015). https://doi.org/10.1007/978-3-319-16727-5_15 14. Fabes, E., Mendez, O., Mitrea, M.: Boundary layers on Sobolev–Besov spaces and Poisson’s equation for the Laplacian in Lipschitz domains. J. Funct. Anal. 159, 323–368 (1998) 15. Grisvard, P.: Elliptic Problems in Nonsmooth Domains. Pitman, Boston (1985) 16. Grzhibovskis, R., Mikhailov, S., Rjasanow, S.: Numerics of boundary-domain integral and integro-differential equations for BVP with variable coefficient in 3D. Comput. Mech. 51, 495–503 (2013). https://doi.org/10.1007/s00466-012-0777-8 17. Hackbusch, W.: Integral Equations: Theory and Numerical Treatment. International Series of Numerical Mathematics, vol. 120. Birkhäuser, Basel (1995) 18. Haroske, D., Triebel, H.: Distributions, Sobolev Spaces, Elliptic Equations. EMS Textbooks in Mathematics. Eur. Math. Soc., Zürich (2008) 19. Hellwig, G.: Partial Differential Equations: An Introduction. Teubner, Stuttgart (1977) 20. Hilbert, D.: Grundzüge Einer Allgemeinen Theorie der Linearen Integralgleichungen, 2nd edn. Teubner, Leipzig (1924) 21. Hsiao, G.C., Wendland, W.L.: Boundary Integral Equations, Springer, Berlin (2008) 22. Jerison, D.S., Kenig, C.E.: The Neumann problem on Lipschitz domains. Bull., New Ser., Am. Math. Soc. 4, 203–207 (1981) 23. Jerison, D.S., Kenig, C.E.: The Dirichlet problem in non-smooth domains. Ann. Math. 113, 367–382 (1981) 24. Jerison, D.S., Kenig, C.E.: Boundary value problems on Lipschitz domains. In: Littman, W. (ed.) Studies in Partial Differential Equations, pp. 1–68. Math. Assoc. of America, Washington (1982) 25. Levi, E.E.: I problemi dei valori al contorno per le equazioni lineari totalmente ellittiche alle derivate parziali. Mem. Soc. Ital. dei Sc. XL 16, 1–112 (1909) 26. Lions, J.-L., Magenes, E.: Non-Homogeneous Boundary Value Problems and Applications, vol. 1. Springer, Berlin (1972) 27. McLean, W.: Strongly Elliptic Systems and Boundary Integral Equations. Cambridge University Press, Cambridge (2000) 28. Mikhailov, S.E.: Finite-dimensional perturbations of linear operators and some applications to boundary integral equations. Eng. Anal. Bound. Elem. 23, 805–813 (1999) 29. Mikhailov, S.E.: Localized boundary-domain integral formulations for problems with variable coefficients. Eng. Anal. Bound. Elem. 26, 681–690 (2002) 30. Mikhailov, S.E.: Analysis of united boundary-domain integro-differential and integral equations for a mixed BVP with variable coefficient. Math. Methods Appl. Sci. 29, 715–739 (2006) 31. Mikhailov, S.E.: Traces, extensions and co-normal derivatives for elliptic systems on Lipschitz domains. J. Math. Anal. Appl. 378, 324–342 (2011). https://doi.org/10.1016/j.jmaa.2010.12.027 32. Mikhailov, S.E.: Solution regularity and co-normal derivatives for elliptic systems with non-smooth coefficients on Lipschitz domains. J. Math. Anal. Appl. 400(1), 48–67 (2013). https://doi.org/10.1016/j.jmaa.2012.10.045 33. Mikhailov, S.E.: Analysis of segregated boundary-domain integral equations for variable-coefficient Dirichlet and Neumann problems with general data. (2015) arXiv:1509.03501 34. Mikhailov, S.E., Mohamed, N.A.: Numerical solution and spectrum of boundary-domain integral equation for the Neumann BVP with variable coefficient. Int. J. Comput. Math. 89(11), 1488–1503 (2012). https://doi.org/10.1080/00207160.2012.679733 35. Mikhailov, S.E., Nakhova, I.S.: Mesh-based numerical implementation of the localized boundary-domain integral equation method to a variable-coefficient Neumann problem. J. Eng. Math. 51, 251–259 (2005) 36. Mikhailov, S.E., Portillo, C.F.: BDIE system to the mixed BVP for the Stokes equations with variable viscosity. In: Constanda, C., Kirsh, A. (eds.) Integral Methods in Science and Engineering: Theoretical and Computational Advances. Springer, Boston (2015). https://doi.org/10.1007/978-3-319-16727-5_33 37. Mikhailov, S.E., Portillo, C.F.: A new family of boundary-domain integral equations for a mixed elliptic BVP with variable coefficient. In: Harris, P. (ed.) Proceedings of the 10th UK Conference on Boundary Integral Methods, pp. 76–84. University of Brighton, Brighton (2015) 38. Miranda, C.: Partial Differential Equations of Elliptic Type, 2nd edn. Springer, Berlin (1970) 39. Mitrea, D.: The method of layer potentials for non-smooth domain with arbitrary topology. Integral Equ. Oper. Theory 29, 320–338 (1997) 40. Mitrea, I., Mitrea, M.: Multy-Layer Potentials and Boundary Problems for Higher-Order Elliptic Systems in Lipschitz Domains. Lecture Notes in Mathematics, vol. 2063. Springer, Berlin (2013) 41. Mitrea, M., Monniaux, S.: The regularity of the Stokes operator and the Fujita–Kato approach to the Navier–Stokes initial value problem in Lipschitz domains. J. Funct. Anal. 254, 1522–1574 (2008) 42. Mitrea, M., Wright, M.: Boundary Value Problems for the Stokes System in Arbitrary Lipschitz Domains. Astérisque, vol. 344. Société Mathématique de France, Paris (2012) 43. Pomp, A.: The Boundary-Domain Integral Method for Elliptic Systems. With Applications in Shells. Lecture Notes in Mathematics, vol. 1683. Springer, Berlin (1998) 44. Pomp, A.: Levi functions for linear elliptic systems with variable coefficients including shell equations. Comput. Mech. 22, 93–99 (1998) Mikhailov Boundary Value Problems (2018) 2018:87 Page 52 of 52 45. Runst, T., Sickel, W.: Sobolev Spaces of Fractional Order, Nemytskij Operators, and Nonlinear Partial Differential Equations. de Gruyter, Berlin (1996) 46. Sladek, J., Sladek, V., Atluri, S.N.: Local boundary integral equation (LBIE) method for solving problems of elasticity with nonhomogeneous material properties. Comput. Mech. 24, 456–462 (2000) 47. Sladek, J., Sladek, V., Zhang, J.-D.: Local integro-differential equations with domain elements for the numerical solution of partial differential equations with variable coefficients. J. Eng. Math. 51, 261–282 (2005) 48. Taigbenu, A.E.: The Green Element Method. Kluwer Academic, Boston (1999) 49. Trenogin, V.A.: Functional Analysis. Nauka, Moscow (1980) 50. Vainberg, M.M., Trenogin, V.A.: Theory of Branching of Solutions of Non-Linear Equations. Noordhoff, Leyden (1974) 51. Verchota, G.: Layer potentials and regularity for the Dirichlet problem for Laplace’s equation in Lipschitz domains. J. Funct. Anal. 59(3), 572–611 (1984) 52. Zhu, T., Zhang, J.-D., Atluri, S.N.: A local boundary integral equation (LBIE) method in computational mechanics, and a meshless discretization approach. Comput. Mech. 21, 223–235 (1998) 53. Zhu, T., Zhang, J.-D., Atluri, S.N.: A meshless numerical method based on the local boundary integral equation (LBIE) to solve linear and non-linear boundary value problems. Eng. Anal. Bound. Elem. 23, 375–389 (1999) 54. Zolesio, J.L.: Multiplication dans les espaces de Besov. Proc. R. Soc. Edinb. 78A, 113–117 (1977)

Journal

Boundary Value ProblemsSpringer Journals

Published: May 31, 2018

References

You’re reading a free preview. Subscribe to read the entire article.


DeepDyve is your
personal research library

It’s your single place to instantly
discover and read the research
that matters to you.

Enjoy affordable access to
over 18 million articles from more than
15,000 peer-reviewed journals.

All for just $49/month

Explore the DeepDyve Library

Search

Query the DeepDyve database, plus search all of PubMed and Google Scholar seamlessly

Organize

Save any article or search result from DeepDyve, PubMed, and Google Scholar... all in one place.

Access

Get unlimited, online access to over 18 million full-text articles from more than 15,000 scientific journals.

Your journals are on DeepDyve

Read from thousands of the leading scholarly journals from SpringerNature, Elsevier, Wiley-Blackwell, Oxford University Press and more.

All the latest content is available, no embargo periods.

See the journals in your area

DeepDyve

Freelancer

DeepDyve

Pro

Price

FREE

$49/month
$360/year

Save searches from
Google Scholar,
PubMed

Create lists to
organize your research

Export lists, citations

Read DeepDyve articles

Abstract access only

Unlimited access to over
18 million full-text articles

Print

20 pages / month

PDF Discount

20% off