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

Abstract

We prove the unique solvability for the Poisson and heat equations in non-smooth domains $\Omega \subset {\mathbb{R}}^d$ in weighted Sobolev spaces. The zero Dirichlet boundary condition is considered, and domains are merely assumed to admit the Hardy inequality: $${\int }_{\Omega }|\frac{f(x)}{d(x,\partial \Omega )}{|}^2\mathrm{d}x\le N{\int }_{\Omega }|\nabla f{|}^2\mathrm{d}x,\forall f\in C_c^{\infty }(\Omega ).$$ To describe the boundary behavior of solutions, we introduce a weight system that consists of superharmonic functions and the distance function to the boundary. The results provide separate applications for the following domains: convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, conic domains, and domains $\Omega \subset {\mathbb{R}}^d$ which the Aikawa dimension of ${\Omega }^c$ is less than $d-2$.