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 Ω⊂ Rᵈ, with the zero Dirichlet boundary condition. Our sole assumption for the domain Ω is the Hardy inequality: There exists a constant $N>0$ such that ∫Ω|f(x)/d(x,∂Ω)|²d\,x≤ N∫Ω|∇ f|² d\,x any f∈ Cc∞(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 Ωᵈ for which the Aikawa dimension of Ωᶜ 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.