Solvability of the divergence equation implies John via Poincar\'e inequality

TL;DR

This paper establishes that the solvability of the divergence equation in Sobolev spaces on a domain is equivalent to the domain being a John domain, characterized by a weighted Poincaré inequality, with extensions to higher dimensions.

Contribution

It proves the equivalence between divergence equation solvability, the John domain condition, and a weighted Poincaré inequality, extending results to higher dimensions under certain assumptions.

Findings

Divergence equation solvability characterizes John domains.

Weighted Poincaré inequality holds if and only if the domain is John.

Results extend to higher dimensions with additional domain conditions.

Abstract

Let $\Omega \subset \rr^2$ be a bounded simply connected domain. We show that, for a fixed (every) $p\in (1,\fz),$ the divergence equation $\mathrm{div}\,\mathbf{v}=f$ is solvable in $W^{1,p}_0(\Omega)^2$ for every $f\in L^p_0(\Omega)$, if and only if $\Omega$ is a John domain, if and only if the weighted Poincar\'e inequality $$\int_\Omega|u(x)-u_{\Omega}|^q\,dx\le C\int_\Omega|\nabla u(x)|^q\dist(x,\partial \Omega)^q\,dx$$ holds for some (every) $q\in [1,\fz)$. In higher dimensions similar results are proved under some additional assumptions on the domain in question.