Trace-free Korn inequalities in Orlicz spaces

TL;DR

This paper establishes necessary and sufficient conditions for Korn inequalities in Orlicz spaces, addressing their validity for various gradient norms and related operators, thus advancing the theoretical understanding of these inequalities.

Contribution

It provides the first comprehensive characterization of Korn inequalities in Orlicz spaces, including conditions for the full symmetric gradient and related operators.

Findings

Conditions for Korn inequalities in Orlicz spaces are fully characterized.

The necessity of these conditions is confirmed for related inequalities.

Results apply to negative Orlicz-Sobolev norms and the Bogovskii operator.

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 Bogovskii operator.