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Divergence operator and Poincaré inequalities on arbitrary bounded domains. (English) Zbl 1205.35044

Summary: Let \(\Omega \) be an arbitrary bounded domain of \(\mathbb R^n\). 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.

MSC:

35F15 Boundary value problems for linear first-order PDEs
35A23 Inequalities applied to PDEs involving derivatives, differential and integral operators, or integrals
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