Higher order Poisson Kernels and $L^p$ polyharmonic boundary value problems in Lipschitz domains

TL;DR

This paper develops higher order Poisson kernels and polyharmonic fundamental solutions to solve boundary value problems in Lipschitz domains, extending classical potential theory to higher orders and less regular domains.

Contribution

It introduces higher order conjugate Poisson kernels and polyharmonic fundamental solutions, enabling solutions to polyharmonic boundary value problems with $L^p$ data in Lipschitz domains.

Findings

Constructed higher order kernels and fundamental solutions.

Provided integral representations for boundary value problems.

Solved Dirichlet, Neumann, and regularity problems in Lipschitz domains.

Abstract

In this article, we introduce higher order conjugate Poisson and Poisson kernels, which are higher order analogues of the classical conjugate Poisson and Poisson kernels, as well as the polyharmonic fundamental solutions, and define multi-layer potentials in terms of Poisson field and the polyharmonic fundamental solutions, in which the former formed by the higher order conjugate Poisson and Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems (i.e., Dirichlet, Neumann and regularity problems) with $L^{p}$ boundary data for polyharmonic equations in Lipschitz domains and give integral representation (or potential) solutions of these problems.