ISSN 0012-2661, Differential Equations, 2017, Vol. 53, No. 7, pp. 891–899.
Pleiades Publishing, Ltd., 2017.
Original Russian Text
S.E. Kholodovskii, 2017, published in Differentsial’nye Uravneniya, 2017, Vol. 53, No. 7, pp. 919–926.
PARTIAL DIFFERENTIAL EQUATIONS
Solution of Boundary Value Problems for the Laplace
Equation in a Ball Bounded by a Multilayer Film
S. E. Kholodovskii
Transbaikal State University, Chita, 672039 Russia
Received October 27, 2016
Abstract—We derive boundary conditions on multilayer ﬁlms bounding a ball and consisting of
inﬁnitely thin strongly and weakly permeable layers and obtain formulas expressing the solutions
of boundary value problems for the Laplace equation in a ball bounded with two-layer ﬁlms by
single quadratures via the solutions of the classical Dirichlet and Neumann problems for the
Laplace equation in the ball (without the ﬁlms).
Multilayer ﬁlm structures (thermal insulators, nanocoatings, screens, membranes, drains, etc.)
are widely used when solving various applied problems. In the monographs [1, p. 110; 2, pp. 349,
352, 406], the solution of problems outside a ﬁlm was sought in the form of Cauchy type integrals
over the boundary of the ﬁlm, and transmission conditions were assumed to be satisﬁed on the
ﬁlm itself. The problems were thus reduced to singular integral equations. The general problem
for a strongly permeable ﬁlm (crack) was reduced to an integro-diﬀerential equation [1, p. 111].
A method in which the desired solutions are represented as some potentials with unknown density
was developed in [3–8]. The case of a strongly permeable ﬁlm for electrostatic problems was con-
sidered in , where these problems were reduced to nonstandard variational problems. The Mellin
integral transform was used to reduce a heat problem in a wedge-shaped domain with a weakly
permeable ﬁlm to a ﬁnite-diﬀerence equation in . The existence and uniqueness of a generalized
solution of diﬀusion problems with a three-layer ﬁlm consisting of a strongly permeable layer with
adjacent weakly permeable layers was proved in , where, to take into account the transmission
conditions on the ﬁlm, distribution coeﬃcients were introduced in the equation. The paper 
dealt with a problem in a half-space bounded by a thin movable ﬁlm. The Hankel and Laplace
integral transforms were used to solve this problem, and the hypothesis that the ﬁlm and its bound-
ary have the same temperature was accepted. The case of multilayer ﬁlms on the boundary of the
half-space (x<0) × (y ∈
) was considered for a class of linear equations in .
1. BOUNDARY CONDITIONS ON MULTILAYER SPHERICAL FILMS:
STATEMENT OF THE PROBLEM
Consider the ball D =(0≤ r<1) × (0 ≤ θ ≤ π) × (0 ≤ σ<2π) whose boundary r =1
is a multilayer ﬁlm of arbitrarily many strongly and weakly permeable layers. [Here (r, θ, σ)are
spherical coordinates.] Following [13, 14], we model strongly (respectively, weakly) permeable
layers by inﬁnitely thin layers with inﬁnite (respectively, inﬁnitesimal) permeability. Here we use
the language of heat and mass transfer problems. Consider the equation r
Δu = 0 for a function
u(r, θ, σ) in the ball D; i.e., we deal with the equation
u)+Lu =0, 0 <r<1, (1)