JournalszaaVol. 31, No. 3pp. 335–356

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

  • Dominic Breit

    Universität des Saarlandes, Saarbrücken, Germany
  • Oliver D. Schirra

    Universität des Saarlandes, Saarbrücken, Germany
Korn-Type Inequalities in Orlicz-Sobolev Spaces Involving the Trace-Free Part of the Symmetric Gradient and Applications to Regularity Theory cover
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Abstract

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

Ωh(v)dxcΩh(EDv)dx\int_\Omega h(|\nabla v|)dx \leqslant c\int_\Omega h(|\mathcal E^D v|)dx

for functions with zero trace as well as some further variants of this inequality. Here,~hh is an NN-function of rather general type. As an application we prove partial C1,αC^{1,\alpha}-regularity of minimizers of energies of the type Ωh(EDv)dx,\int_\Omega h(|\mathcal E^D v|)dx, occurring, for example, in general relativity.

Cite this article

Dominic Breit, Oliver D. Schirra, Korn-Type Inequalities in Orlicz-Sobolev Spaces Involving the Trace-Free Part of the Symmetric Gradient and Applications to Regularity Theory. Z. Anal. Anwend. 31 (2012), no. 3, pp. 335–356

DOI 10.4171/ZAA/1463