Weighted Sobolev space theory for Poisson's equation in non-smooth domains
Jinsol Seo · arXiv (Cornell University) · 2024
We introduce a general $L_p$-solvability result for the Poisson equation in non-smooth domains $Ω\subset \mathbb{R}^d$, with the zero Dirichlet boundary condition. Our sole assumption on the domain $Ω$ is the Hardy inequality: There exists a constant $N>0$ such that $$ \int_Ω\Big|\frac{f(x)}{d(x,\partialΩ)}\Big|^2\,\mathrm{d} x\leq N\int_Ω| abla f|^2 \,\mathrm{d} x\quad\text{for any}\quad f\in C_c^{\infty}(Ω)\,. $$ 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 $Ω\subset\mathbb{R}^d$ for which the Aikawa dimension of $Ω^c$ 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 Hölder continuity of solutions.