Green function representation for the Dirichlet problem of the polyharmonic equation in a sphere

Abstract

The representation of solutions in an explicit form for boundary value problems of differential equations is always very important. In this article the Green function for the Dirichlet problem is constructed in an explicit form inside a sphere for the polyharmonic equations. We note that the obtained formulas have an independent sense. In particular, the explicit representation of the solution of the Dirichlet problem for the polyharmonic equation has important consequences in the theory of elasticity.

About solvability of boundary value problems for the nonhomogeneous polyharmonic equation in a ball

“…with homogeneous boundary conditions In this case the rank of the matrix A 0 D A.k 1 ; k 2 ; : : : ; k m / is m. It is known that problem (3.1)-(3.2) is uniquely solvable for any right-hand side, and the solution is represented by the Green function [7,14,15]. As a result we constructed the Green function in explicit form.…”

On a biharmonic problem in a circular ring

“…Explicit formula for the polyharmonic Green function in the unit ball of R n was obtained recently by Kalmenov, Koshanov and Nemchenko, see [14].…”

On Green’s Function of the Dirichlet Problem for the Polyharmonic Equation in the Ball

“…[2,3], and the Green's functions in the sector for the biharmonic and 3-harmonic equations are presented in [4,5]. The papers [6][7][8] are devoted to the construction of the Green's function to the Dirichlet problem for the polyharmonic equation in the unit ball, and the paper [9] contains solutions to the Dirichlet and Neumann [10] problems for the homogeneous polyharmonic equation. The papers [11,12] give an explicit form of the Green's functions for the biharmonic and 3-harmonic equations in the unit ball.…”