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

TL;DR

This paper proves that viscosity solutions to a class of Aronsson's equations are differentiable everywhere in their domain, extending previous results on infinity harmonic functions to more general elliptic equations.

Contribution

It establishes the everywhere differentiability of viscosity solutions for a broader class of Aronsson's equations with uniformly elliptic coefficients.

Findings

Viscosity solutions are differentiable everywhere in the domain.

Extension of differentiability results from infinity harmonic functions to general Aronsson's equations.

Applicable to equations with coefficients in C^{1,1} class, broadening previous scope.

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 $$ <D_x(H(x, Du)), D_p H(x, Du)>=0 \quad \rm in\ \ \Omega, $$ where $H$ is given by $$H(x,\,p)=<A(x)p,p>=\sum_{i,\,j=1}^na^{ij}(x)p_i p_j,\ x\in\Omega, \ p\in\mathbb R^n, $$ 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.