New Korn-type inequalities and regularity of solutions to linear elliptic systems and anisotropic variational problems involving the trace-free part of the symmetric gradient

Schirra, Oliver
January 2012
Calculus of Variations & Partial Differential Equations;Jan2012, Vol. 43 Issue 1/2, p147
Academic Journal
The aim of this note is to investigate a regularity theory for minimizers of energies whose density depends on the trace-free part of the symmetric gradient, where integrands of anisotropic growth are considered. An adequate coercive inequality guarantees the existence of minimizers of such energies in suitable Sobolev classes. Moreover, various other Korn-type inequalities are shown, which can be used to prove the smoothness of weak solutions to linear elliptic systems involving the trace-free part of the symmetric gradient. In particular, Campanato-type estimates for solutions to such systems are established so that all tools are available to prove the interior regularity of minimizers of energies depending on the trace-free part of the symmetric gradient.


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