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We consider solutions $u\in W^{1,p}\big(\Omega;\mathbb{R}^{N}\big)$ of the$p$-Laplacian PDE \begin{equation}\nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for$x\in\Omega\subseteq\mathbb{R}^{n}$, where $\Omega$ is open and bounded. Moregenerally, 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} Ineach case, the coefficient map $a\ : \ \Omega\rightarrow\mathbb{R}$ is onlyassumed to be of class $VMO(\Omega)\cap L^{\infty}(\Omega)$, which means thatit may be discontinuous. Without assuming that $x\mapsto a(x)$ has any weakdifferentiability, we show that$u\in\mathscr{C}_{\text{loc}}^{0,\alpha}(\Omega)$ for each $0<\alpha<1$. Thepreceding 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 onlyasymptotically convex.

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