Weighted $L^p$ Boundary Value Problems for Laplace's Equation on (Semi-)Convex Domains

Abstract

Let $n \geq 2$ and $\Omega$ be a bounded (semi-)convex domain in $\mathbb{R}^n$. Assume that $p \in (1,\infty)$ and $\omega \in A_p(\partial \Omega)$, where $A_p(\partial \Omega)$ denotes the Muckenhoupt weight class on $\partial \Omega$, the boundary of $\Omega$. In this article, the author proves that the Dirichlet and Neumann problems for Laplace's equation on $\Omega$ with boundary data in the weighted space $L^p_{\omega}(\partial \Omega)$ are uniquely solvable. Moreover, the unique solvability of the Regularity problem for Laplace's equation on $\Omega$ with boundary data in the weighted Sobolev space $\dot{W}^p_{1,\omega}(\partial \Omega)$ is also obtained. Furthermore, the weighted $L^p_{\omega}(\partial \Omega)$-estimates for the Dirichlet, Regularity and Neumann problems are established.