Neumann Type Problems for the Polyharmonic Equation in Ball
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Journal of Mathematical Sciences, Vol. 249, No. 6, September, 2020
NEUMANN TYPE PROBLEMS FOR THE POLYHARMONIC EQUATION IN BALL V. V. Karachik South Ural State University 76, Pr. Lenina, Chelyabinsk 454080, Russia [email protected]
UDC 517.956.223+517.575
For Neumann type problems for the homogeneous polyharmonic equation in the unit ball we obtain necessary solvability conditions in the form of orthogonality of homogeneous harmonic polynomials to linear combinations of boundary functions with coefficients taken from the integer Neumann triangle. Bibliography: 18 titles.
1
Introduction
The polyharmonic equation is an important representative of linear elliptic higher order partial differential equations. The solvability conditions for the Dirichlet (cf., for example, [1, 2]) and Neumann (cf., for example, [3]–[5]) problems for the polyharmonic equation are studied in the classical theory of boundary value problems for elliptic equations satisfying the complementarity condition. A more general boundary value problem for the polyharmonic equation was considered in [6], where the boundary conditions contain higher degree polynomials of normal derivatives. The solvability theorem and a representation formula for the solution were obtained there. Solvability conditions for boundary value problems for the polyharmonic equation in the ball with normal derivatives in the boundary conditions were also studied in [7]. In the cited works, the solvability conditions are formulated as the orthogonality conditions for some vector-valued functions depending on the data of the problem or the equality of ranks of higher order special matrices. To determine under what boundary conditions a particular problem is solvable, complicated calculations are required. In this paper, we study a class of boundary value problems which is a natural generalization of the classical Neumann problem for the polyharmonic equation [3]. Therefore, it is reasonable to find easily verifiable solvability conditions. This paper is devoted to obtaining such solvability conditions for Neumann type problems. Section 2 contains auxiliary notions and assertions (Lemmas 2.1 and 2.2) necessary to formulate and prove the main result presented by Theorem 3.1. In Section 4, we provide illustrative examples.
Translated from Problemy Matematicheskogo Analiza 103, 2020, pp. 143-154. c 2020 Springer Science+Business Media, LLC 1072-3374/20/2496-0974
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We consider the unit ball S ={x ∈ Rn : |x| < 1} in Rn and the unit sphere ∂S = {x ∈ Rn : |x| = 1}, where |x| = x21 + x22 + . . . + x2n . In the unit ball S, we consider the class of Neumann type boundary value problems (Problems Nk , k ∈ N) for the homogeneous polyharmonic equation Δm u = 0,
x ∈ S,
∂ k u = ϕ1 (s), ∂ν k ∂S
(1.1) ∂ k+1 u ∂ k+m−1 u = ϕ (s), . . . , = ϕm (s), 2 ∂ν k+1 ∂S ∂ν k+m−1 ∂S
s ∈ ∂S,
(1.2)
∂ is the outward normal derivative relative to the unit sphere and the functions ϕi , where ∂ν i = 1, . . . , m, are defined on ∂S and are smooth as specified below. The class of Problems Nk
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