Higher order Poisson kernels and Lp polyharmonic boundary value 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 the Poisson field and the polyharmonic fundamental solutions, in which the former is formed by the higher order conjugate Poisson and the Poisson kernels. Then by the multi-layer potentials, we solve three classes of boundary value problems (i.e., Dirichlet, Neumann and regularity problems) with Lp boundary data for polyharmonic equations in Lipschitz domains and give integral representation (or potential) solutions of these problems.