Holder continuity of weak solutions of p-Laplacian PDE's with VMO coefficients

We consider solutions $u\in W^{1,p}(\mathrm{\Omega };{\mathbb{R}}^N)$ of the $p$-Laplacian PDE \begin{equation} \nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for $x\in \mathrm{\Omega }\subseteq {\mathbb{R}}^n$, where $\mathrm{\Omega }$ is open and bounded. More generally, we consider solutions of the elliptic system \begin{equation} \nabla\cdot\left(a(x)g’\big(a(x)|Du|\big)\frac{Du}{|Du|}\right)=0\text{, }x\in\Omega\notag \end{equation} as well as minimizers of the functional \begin{equation} \int_{\Omega}g\big(a(x)|Du|\big)\ dx.\notag \end{equation} In each case, the coefficient map $a:\mathrm{\Omega }\to \mathbb{R}$ is only assumed to be of class $VMO(\mathrm{\Omega })\cap L^{\mathrm{\infty }}(\mathrm{\Omega })$, which means that it may be discontinuous. Without assuming that $x\mapsto a(x)$ has any weak differentiability, we show that $u\in {\mathcal{C}}_{\text{loc}}^{0,\alpha }(\mathrm{\Omega })$ for each $0<\alpha <1$. The preceding results are, in fact, a corollary of a much more general result, which applies to the functional \begin{equation} \int_{\Omega}f\big(x,u,Du\big)\ dx\notag \end{equation} in case $f$ is only asymptotically convex.