Korn-type inequalities in orlicz-sobolev spaces involving the trace-free part of the symmetric gradient and applications to regularity theory

We prove variants of Korn's inequality involving the trace-free part of the symmetric gradient of vector fields v : Omega -> R-n (Omega subset of R-n), that is,

f(Omega)(h(vertical bar del nu vertical bar) (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 regularity of of minimizers of energies of the type f(Omega) h(vertical bar epsilon(D)(v)vertical bar) dx, occurring, for example, in general relativity.