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We prove variants of Korn’s inequality involving the deviatoric part of the symmetric gradient of fields $$ u:{\mathbb{R}^2} \supset \Omega \to {\mathbb{R}^2} $$ belonging to Orlicz–Sobolev classes. These inequalities are derived with the help of gradient estimates for the Poisson equation in Orlicz spaces. We apply these Korn type inequalities to variational integrals of the form     $$ \int\limits_\Omega {h\left( {\left| {{\varepsilon^D}(u)} \right|} \right)dx} $$    occurring in General Relativity and prove C 1,α–regularity results for minimizers under rather general conditions on the N–function h. A further useful tool for this analysis is an appropriate version of the (Sobolev-) Poincaré inequality with ε D (u) measuring the distance of u to the holomorphic functions. Bibliography: 20 titles.

journal of mathematical sciences

Generalizations of Korn’s inequality based on gradient estimates in Orlicz spaces and applications to variational problems in 2D involving the trace free part of the symmetric gradient