Korn's inequality and John domains

TL;DR

This paper proves that in simply connected planar domains, Korn's inequality holds if and only if the domain is a John domain, establishing an equivalence with other fundamental inequalities.

Contribution

It establishes the converse of Korn's inequality on domains satisfying a separation condition, characterizing John domains via inequality equivalences.

Findings

Korn's inequality holds iff the domain is John in simply connected planar domains

Equivalence of Korn's, Babuška-Aziz, and Friedrich's inequalities in these domains

Characterization of domain geometry through inequality conditions

Abstract

It is quite well known that Korn's inequality is true on all John domains. We are interested in the converse implication under assumption of so called separation condition of the domain. Our result implies that in a simply connected planar domain the Korn's inequality holds if and only if the domain is John. In particular, we obtain the equivalence of Korn's inequality, Babu\v ska-Aziz inequality and Friedrich's inequality in simply connected planar domains.