Sobolev space theory for Poisson's equation in non-smooth domains via superharmonic functions and Hardy's inequality

We introduce a general Lₚ-solvability result for the Poisson equation in non-smooth domains Rᵈ, with the zero Dirichlet boundary condition. Our sole assumption for the domain is the Hardy inequality: There exists a constant N>0 such that _|f (x) d (x, ) |²d\, x N_| f|² d\, x any f Cc^ () \,. 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 ᵈ for which the Aikawa dimension of ᶜ is less than d-2. Using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted Lₚ-solvability results for various non-smooth domains and specific weight ranges that differ for each domain condition. Furthermore, we provide an application for the H\"older continuity of solutions.