Partial regularity for minimizers of discontinuous quasi-convex integrals with degeneracy

In this paper we are concerned with the regularity of minimizers u is an element of W-1,W-p(Omega, R-N) of quasi-convex integral functionals of the type F[u] := integral(Omega) f(x, u, Du)dx. The crucial point here is that the integrand f admits very weak regularity properties. With respect to the gradient variable it satisfies degenerate/singular p-growth conditions without necessarily possessing a quasi-diagonal Uhlenbeck-type structure, and with respect to the x-variable the integrand might be even discontinuous. It is only assumed that a certain VMO-condition holds. Under those assumptions we prove partial Holder continuity of minimizers, i.e. Holder continuity of u for any Holder exponent alpha is an element of (0,1) outside a set of measure zero. Under such weak assumptions regularity results for the gradient of minimizers is not expected to hold since even in the scalar case counterexamples to C-1-regularity are known. (C) 2011 Elsevier Inc. All rights reserved.