Polynomial solutions of the boundary value problems for the Poisson equation in a layer

TL;DR

This paper proves that for Poisson equations in a layered domain with polynomial data, the solutions under certain boundary conditions are uniquely polynomial, and provides an algorithm for constructing these solutions.

Contribution

It establishes the polynomial nature and uniqueness of solutions for boundary value problems with polynomial data in layered domains, and introduces a constructive algorithm.

Findings

Solutions are unique and polynomial under specified conditions.

An explicit algorithm for constructing polynomial solutions is provided.

Examples illustrate the solution process.

Abstract

In a multidimensional infinite layer bounded by two hyperplanes, the Poisson equation with the polynomial right-hand side is considered. It is shown that the Dirichlet boundary value problem and the mixed Dirichlet-Neumann boundary value problem with polynomial boundary conditions have a unique solution in the class of functions of polynomial growth and it solution is a polynomial. An algorithm for constructing this polynomial solution is given and examples are considered.