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

Abstract

We show the everywhere differentiability of viscosity solutions to a class of Aronsson equations in ${\mathbb{R}}^n$ for $n\ge 2$, where the coefficient matrices $A$ are assumed to be uniformly elliptic and $C^{1,1}$. Our result extends an earlier important theorem by Evans and Smart [18] who have studied the case $A=I_n$ which correspond to the $\infty$-Laplace equation. We also show that every point is a Lebesgue point for the gradient.

In the process of proving the results we improve some of the gradient estimates obtained for the infinity harmonic functions. The lack of suitable gradient estimates has been a major obstacle for solving the $C^{1,\alpha }$ problem in this setting, and we aim to take a step towards better understanding of this problem, too.

A key tool in our approach is to study the problem in a suitable intrinsic geometry induced by the coefficient matrix $A$. Heuristically, this corresponds to considering the question on a Riemannian manifold whose the metric is given by the matrix $A$.