Solvability of the divergence equation implies John via Poincaré inequality

Abstract

Let $Ω\subset \rr^2$ 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 $W_0^{1,p}(\Omega {)}^2$ for every $f\in L_0^p(\Omega )$, if and only if $\Omega$ is a John domain, if and only if the weighted Poincaré inequality $$\int_Ω|u(x)-u_Ω|^q\,dx\le C\int_Ω|\nabla u(x)|^q\dist(x,\partial Ω)^q\,dx$$ holds for some (every) $q\in [1,\fz)$. In higher dimensions similar results are proved under some additional assumptions on the domain in question.