Trace-Free Korn Inequalities in Orlicz Spaces

Abstract

Necessary and sufficient conditions are exhibited for a Korn-type inequality to hold between (possibly different) Orlicz norms of the gradient of vector-valued functions and of the deviatoric part of their symmetric gradients. As a byproduct of our approach, a positive answer is given to the question of the necessity of the same sufficient conditions in related Korn-type inequalities for the full symmetric gradient, for negative Orlicz--Sobolev norms, and for the gradient of the Bogovskiǐ operator.