Holder continuity of weak solutions of p-Laplacian PDEs with VMO coefficients

We consider solutions u is an element of W-1,W-p(Omega; R-N) of the p-Laplacian PDE del . (a(x)vertical bar Du vertical bar(p-2) Du) = 0, for x is an element of Omega subset of R-n, where Omega is open and bounded. More generally, we consider solutions of the elliptic system del . (a(x)g' (a(x)vertical bar Du vertical bar)Du/vertical bar Du vertical bar) = 0,x is an element of Omega as well as minimizers of the functional integral(g)(Omega)(a(x)vertical bar Du vertical bar) dx. In each case, the coefficient map a : Omega -> R is only assumed to be of class VMO(Omega) boolean AND L-infinity (Omega), which means that it may be discontinuous. Without assuming that x bar right arrow a(x) has any weak differentiability, we show that u is an element of C-loc(0,alpha) (Omega) for each 0 < alpha < 1. The preceding results are, in fact, a corollary of a much more general result, which applies to the functional integral(f)(Omega)(x, u, Du) dx in case f is only asymptotically convex. (C) 2019 Elsevier Ltd. All rights reserved.