Paper
Weighted Sobolev space theory for Poisson's equation in non-smooth domains
Authors
Jinsol Seo
Abstract
We introduce a general -solvability result for the Poisson equation in non-smooth domains , with the zero Dirichlet boundary condition. Our sole assumption on the domain is the Hardy inequality: There exists a constant such that 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 for which the Aikawa dimension of is less than . Using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted -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.