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

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

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