Everywhere differentiability of viscosity solutions to a class of Aronsson's equations

Abstract

For any open set $\Omega\subset\mathbb R^n$ and $n\ge 2$, we establish everywhere differentiability of viscosity solutions to the Aronsson equation $$=0 \quad \rm in\ \ \Omega,$$ where $H$ is given by $$H(x,\,p)==\sum_{i,\,j=1}^na^{ij}(x)p_i p_j,\ x\in\Omega, \ p\in\mathbb R^n,$$ and $A=(a^{ij}(x))\in C^{1,1}(\bar\Omega,\mathbb R^{n\times n})$ is uniformly elliptic. This extends an earlier theorem by Evans and Smart \cite{es11a} on infinity harmonic functions.Comment: 24 page