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

Abstract

We consider solutions $u\in W^{1,p}(\Omega ;{\mathbb{R}}^N)$ of the $p$-Laplacian PDE \begin{equation} \nabla\cdot\big(a(x)|Du|^{p-2}Du\big)=0,\notag \end{equation} for $x\in \Omega \subseteq {\mathbb{R}}^n$, 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\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 $a:\Omega \to \mathbb{R}$ 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 $u\in {\mathcal{C}}_{\text{loc}}^{0,\alpha }(\Omega )$ 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_Ωf\big(x,u,Du\big)\ dx\notag \end{equation} in case $f$ is only asymptotically convex.

Nonlinear Analysis

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