Korn-Type Inequalities in Orlicz-Sobolev Spaces Involving the Trace-Free Part of the Symmetric Gradient and Applications to Regularity Theory

Abstract

We prove variants of Korn's inequality involving the trace-free part of the symmetric gradient of vector fields $v:\Omega \to {\mathbb{R}}^n$ ($\Omega \subset {\mathbb{R}}^n$), that is,

for functions with zero trace as well as some further variants of this inequality. Here,~$h$ is an $N$-function of rather general type. As an application we prove partial $C^{1,\alpha }$-regularity of minimizers of energies of the type ${\int }_{\Omega }h(|{\mathcal{E}}^{D}v|)dx,$ occurring, for example, in general relativity.