Korn's inequality and John domains

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.