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

Abstract

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. 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 $a\ : \ \Omega\rightarrow\mathbb{R}$ is only assumed to be of class $VMO(\Omega)\cap L^{\infty}(\Omega)$, which means that it may be discontinuous. Without assuming that $x\mapsto a(x)$ has any weak differentiability, we show that $u\in\mathscr{C}_{\text{loc}}^{0,\alpha}(\Omega)$ for each $0<\alpha<1$. The preceding results are, in fact, a corollary of a much more general result, which applies to the functional \begin{equation} \int_{\Omega}f\big(x,u,Du\big)\ dx\notag \end{equation} in case $f$ is only asymptotically convex.