Divergence operator and Poincare inequalities on arbitrary bounded domains

Details Divergence operator and Poincare inequalities on arbitrary bounded domains Divergence operator and Poincare inequalities on arbitrary bounded domains

2009-06-11

arxiv.org/abs/0906.2050v1

Mathematics

Analysis of PDEs

Scientific paper

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 Poincar\'e 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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