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Let $\Omega \subset \rr^2$ be a bounded simply connected domain. We showthat, 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 ifthe weighted Poincar\'e inequality $$\int_\Omega|u(x)-u_{\Omega}|^q\,dx\leC\int_\Omega|\nabla u(x)|^q\dist(x,\partial \Omega)^q\,dx$$ holds for some(every) $q\in [1,\fz)$. In higher dimensions similar results are proved undersome additional assumptions on the domain in question.

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