ISSN 1066-369X, Russian Mathematics, 2018, Vol. 62, No. 2, pp. 34–48.
Allerton Press, Inc., 2018.
Original Russian Text
V.V. Karachik, 2018, published in Izvestiya Vysshikh Uchebnykh Zavedenii. Matematika, 2018, No. 2, pp. 39–53.
Solving a Problem of Robin Type for Biharmonic Equation
V. V. Karachi k
South Ural State University
pr. Lenina 76, Chelyabinsk, 454080 Russia
Received October 24, 2016
Abstract—We investigate existence and uniqueness conditions for solution to one Robin type
problem for inhomogeneous biharmonic equation in the unit ball. We construct polynomial solution
to the problem when the boundary functions of the problems are polynomials.
Keywords: biharmonic equation, Robin type problem, solvability conditions, harmonic poly-
nomials, polynomial solutions.
Steady-state processes of diﬀerent physical nature are described by elliptic type equations. One of
important cases of the fourth order elliptic equations is the biharmonic equation Δ
many cases the solving of problems of plane elasticity theory reduces to integration of a biharmonic
equation under appropriate boundary conditions. Many problems of continuum mechanics are reduced
to solving harmonic and biharmonic equations; however useful analytic expressions for solutions
are obtained for certain domains of partial form only. Multiple investigations [1–3] are devoted to
applications of biharmonic problems in mechanics and physics.
For the biharmonic equation, the Dirichlet problem is well-known. Polynomial solutions for this
problem were constructed in . Recently, other boundary-value problems were also studied actively: the
Riquier problem, the Neumann problem , the Roben problem . In the spectral theory, the Steklov
spectral problem is topical .
The resolvability solutions of boundary-value problems for elliptic equations and systems of equations
are reduced to the “complementarity” condition. One established that all problems of this type are
Fredholm ones, therefore their resolvability for homogeneous edge conditions are guaranteed by the
orthogonality of the right-hand sides to all solutions of the homogeneous conjugate equation. In this
case, it is possible to obtain more detailed results.
The present paper continues investigations  and has the following structure. In Section 1 for
a biharmonic equation in the unit ball we formulate the boundary-value problem (1)–(2) with edge
conditions in the general form, which we call the Roben type problem. In Theorem 1 from Section 2
we present uniqueness conditions of solution to the investigated problem. Then we ﬁnd necessary and
suﬃcient conditions of resolvability of problem (1)–(2) for the homogeneous biharmonic equation. In
Theorem 2 of Section 3 we consider the case of problem (1)–(2) for the inhomogeneous biharmonic
equation. Finally, in Section 4, based on Theorems 3 and 4, we construct the polynomial solution to
problem (1)–(2), when boundary functions of the problem are polynomials. The results obtained in
Theorems 1 and 2 are illustrated in Examples 1–4.