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

In this paper we are concerned with the regularity of minimizers u∈W1,p(Ω,RN)u∈W1,p(Ω,RN) of quasi-convex integral functionals of the typeF[u]:=∫Ωf(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 Hölder continuity of minimizers, i.e. Hölder continuity of u   for any Hölder exponent α∈(0,1)α∈(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 C1C1-regularity are known.