Divergence operator and Poincaré inequalities on arbitrary bounded domains
Résumé
Let $\Omega$ be an arbitrary bounded domain of $\R^n$. We study the right invertibility of the divergence on $\Omega$ in weighted Lebesgue and Sobolev spaces on $\Omega$, and rely this invertibility to a geometric characterization of $\Omega$ and to weighted Poincaré inequalities on $\Omega$. We recover, in particular, well-known results on the right invertibility of the divergence in Sobolev spaces when $\Omega$ is Lipschitz or, more generally, when $\Omega$ is a John domain, and focus on the case of $s$-John domains.
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