Partial regularity of minimizers of functionals with discontinuous coefficients of low integrability with applications to nonlinear elliptic systems

In this article, we consider minimizers of the functional integral(Omega)f (a(x)vertical bar Du vertical bar)dx, where Omega subset of R-n, is open and bounded and u : Omega -> R-N. We show that even though the coefficient a satisfies only that a is an element of W-loc(1,q)(Omega) boolean AND L infinity(Omega), it nonetheless follows that u is partially Holder continuous on the set of regular points of Da. We note that the integrability exponent q>1 is arbitrary, and so, the coefficient map x -> a(x) can be very irregular. As an application we show that weak solutions of the elliptic system del . (a(x)f'(a(x)vertical bar Du vertical bar/vertical bar Du vertical bar) = 0, x is an element of Omega satisfies the same partial Holder continuity. The model case we have in mind is del.(a(x)vertical bar Du vertical bar(p-2)Du) = 0, Finally, we also demonstrate that Du is also almost everywhere Holder continuous.