Korn inequality and divergence operator: Counterexamples and optimality of weighted estimates

Abstract

The Korn inequality and related results on solutions of the divergence in Sobolev spaces have been widely studied since the pioneering works by Korn and Friedrichs. In particular, it is known that this inequality is valid for Lipschitz domains as well as for the more general class of John domains. On the other hand, a few known counterexamples show that those results are not valid for certain bounded domains having external cusps. The goal of this paper is to give very simple counterexamples for a class of cuspidal domains in Rn. Moreover, we show that these counterexamples can be used to prove the optimality of recently obtained results involving weighted Sobolev spaces.