An inhomogeneous polyharmonic Dirichlet problem with $L^p$ boundary data in the upper half-plane

TL;DR

This paper addresses the inhomogeneous polyharmonic Dirichlet problem in the upper half-plane with $L^p$ boundary data, providing a unique integral representation solution using advanced kernel and operator techniques.

Contribution

It introduces a novel integral representation for the polyharmonic Dirichlet problem in the upper half-plane utilizing higher order Poisson kernels and Pompeiu operators.

Findings

Established unique solutions under certain estimates.

Extended methods to inhomogeneous boundary data.

Applied advanced integral operators for solution representation.

Abstract

In this paper, it is investigated for an inhomogeneous Dirichlet problem with $L^p$ boundary data for polyharmonic equation in the upper half-plane. By using higher order Poisson kernels and Pompeiu operators, which are respectively due to Du, Qian and Wang [Z. Du, T. Qian and J. Wang, {\it $L^{p}$ polyharmonic Dirichlet problems in regular domains II: The upper half plane}, J. Differential Equations 252(2012), 1789-1812] as well as Begehr and Hile [H. Begehr and G. Hile, {\it A hierarchy of integral operators}, Rocky Mountain J. Math. 27(1997), 669-706], it is given that the unique integral representation solution under some certain estimates.