Divergence operator and Poincare inequalities on arbitrary bounded domains

TL;DR

This paper investigates the invertibility of the divergence operator on arbitrary bounded domains in $ ^n$, linking it to geometric properties of the domain and establishing weighted Poincaré inequalities, extending known results to more general domains.

Contribution

It provides a geometric characterization of domains for divergence invertibility and extends Poincaré inequalities to $s$-John domains, generalizing classical results.

Findings

Invertibility of divergence characterized by domain geometry.

Extension of weighted Poincaré inequalities to $s$-John domains.

Recovery of classical results for Lipschitz and John domains.

Abstract

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.