Green function for analogue of Robin problem for polyharmonic equation

Abstract

We propose a method of constructing the Green function for some boundary value problems for a polyharmonic equation in a multi-dimensional unit ball. The considered problem are analogues of the Robin problem for an inhomogeneous polyharmonic equation. For studying the solvability of these problems in the class of smooth in a ball functions, we first provide the properties of integral-differential operators. Then, employing these properties, the considered problems are reduced to an equivalent Dirichlet problem with a special right hand side. Using then known statements on the Dirichlet problem, for the main problems we prove the unique solvability theorems. We also obtain integral representations for solutions of these problems via the solutions of the Dirichlet problem. Employing the explicit form of the Green function, we find an integral representation of the Dirichlet problem with a special right hand side. The obtained integral representation then is used to construct the Green function for analogues of Robin problems. We also provide an approach for constructing the Green function for other main problems. In order to do this, we study the differential properties of the fundamental solution of the polyharmonic operator. The obtained properties of the fundamental solutions are applied for studying the properties of the Green function for the Dirichlet problem. We construct the representations of the Green function for analogues of the Robin problem. While finding the Green functions for these problems, we employ essentially the form of the Green function for the Dirichlet problem for the polyhgarmonic equation. Namely, the Green function of these problems is represented as the sum of the Green function for the Dirichlet problem and some integral term. The obtained results are in agreement with the known results for the Laplace operator.