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: $$ \int_{\Omega}\Big|\frac{f(x)}{d(x,\partial\Omega)}\Big|^2\,\,\mathrm{d} x\leq N\int_{\Omega}|\nabla f|^2 \,\mathrm{d} x\,\,\,\,,\,\,\,\, \forall f\in C_c^{\infty}(\Omega)\,. $$ 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 $d-2$.