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# Zeitschrift für Analysis und ihre Anwendungen

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**Volume 31, Issue 3, 2012, pp. 335–356**

**DOI: 10.4171/ZAA/1463**

Published online: 2012-07-11

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

Dominic Breit and Oliver D. SchirraWe prove variants of Korn's inequality involving the trace-free part of the symmetric gradient of vector fields \(v:\Omega\rightarrow\mathbb{R}^n\) (\(\Omega\subset\mathbb{R}^n\)), that is, \[ \int_\Omega h(|\nabla v|)dx \leqslant c\int_\Omega h(|\mathcal E^D v|)dx \] for functions with zero trace as well as some further variants of this inequality. Here,~\(h\) 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.

*Keywords: *Generalized Korn inequalities in Orlicz-Sobolev spaces, variational problems, nonstandard growth, regularity

Breit Dominic, Schirra Oliver: 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), 335-356. doi: 10.4171/ZAA/1463