Weighted Sobolev space theory for the heat equation and the time-fractional heat equation in non-smooth domains

TL;DR

This paper develops a unified weighted Sobolev space framework for solving classical and fractional heat equations in various non-smooth domains, extending solvability and regularity results beyond smooth settings.

Contribution

It introduces a general $L_p$-solvability framework using weighted Sobolev spaces tailored to non-smooth domains with Hardy inequalities, covering diverse geometric conditions.

Findings

Established weighted $L_p$-solvability in non-smooth domains.

Derived boundary regularity and pointwise estimates for solutions.

Extended results to fractional heat equations in complex geometries.

Abstract

We present a general $L_p$-solvability framework for both the classical and time-fractional heat equations in non-smooth domains under the zero Dirichlet boundary condition. We consider domains $\Omega$ admitting the Hardy inequality: There exists a constant $N>0$ such that $$ \int_{\Omega}\Big|\frac{f(x)}{d(x,\partial\Omega)}\Big|^2\,\mathrm{d} x\leq N\int_{\Omega}|\nabla f|^2 \,\mathrm{d} x\quad\text{for any}\quad f\in C_c^{\infty}(\Omega)\,. $$ To illustrate the boundary behavior of solutions in a general framework, we employ a weight system composed of a superharmonic function and a distance function to the boundary. Further, we investigate applications to various non-smooth domains, including convex domains, domains with exterior cone condition, totally vanishing exterior Reifenberg domains, and domains $\Omega\subset\mathbb{R}^d$ for which the Aikawa dimension of $\Omega^c$ is less than $d-2$. By using superharmonic functions tailored to the geometric conditions of the domain, we derive weighted $L_p$-solvability results for various non-smooth domains, with specific weight ranges that differ for each domain condition. In addition, we provide an application to the H\"older continuity of solutions in domains with the volume density condition, as well as pointwise estimates for solutions in Lipschitz cones.