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 $$<D_x(H(x,Du)),D_{p}H(x,Du)>=0\mathrm{in}\mathrm{\Omega },$$ where $H$ is given by $$H(x,p)=<A(x)p,p>=\sum _{i,j=1}^n{a}^{ij}(x)p_i{p}_j,x\in \Omega ,p\in {\mathbb{R}}^n,$$ 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.

24 pages

Cited by in corpus (1)

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