Weighted Sobolev space theory for Poisson's equation in non-smooth domains

Abstract

We introduce a general $L_p$-solvability result for the Poisson equation in non-smooth domains $\Omega\subset \mathbb{R}^d$, with the zero Dirichlet boundary condition. Our sole assumption on the domain $\Omega$ is the Hardy inequality: There exists a constant $N>0$ such that $$ \int_{\Omega}\Big|\frac{f(x)}{d(x,\partial\Omega)}\Big|^2\,\mathrm{d} x\leq N\int_{\Omega}|\nabla f|^2 \,\mathrm{d} x\quad\text{for any}\quad f\in C_c^{\infty}(\Omega)\,. $$ To describe the boundary behavior of solutions in a general framework, we propose a weight system composed of a superharmonic function and the distance function to the boundary. Additionally, we explore applications across a variety of non-smooth domains, including convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, and domains $\Omega\subset\mathbb{R}^d$ for which the Aikawa dimension of $\Omega^c$ is less than $d-2$. Using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted $L_p$-solvability results for various non-smooth domains and specific weight ranges that differ for each domain condition. Furthermore, we provide an application to the H\"older continuity of solutions.