Sobolev gradients of viscosity supersolutions

TL;DR

This paper explores the conditions under which viscosity supersolutions of certain elliptic PDEs possess local Sobolev regularity, emphasizing the role of the asymptotic cone of the operator's sublevel set.

Contribution

It characterizes the Sobolev regularity of viscosity supersolutions based on the geometric comparison of the asymptotic cone to a minimal operator's sublevel set.

Findings

The asymptotic cone's shape influences supersolution regularity.

Comparison with a minimal operator determines Sobolev space inclusion.

Conditions for regularity depend on geometric properties of the operator.

Abstract

We investigate which elliptic PDEs that have the property that every viscosity supersolution is $W^{1,q}_{loc}(\Omega)$, $\Omega\subseteq\mathbb{R}^n$. The asymptotic cone of the operator's sublevel set seems to be essential. It turns out that much can be said if we know how this cone compares to the sublevel set of a certain minimal operator associated with the exponent $q$.