A Compactness Condition for the Theorem of Sums in a Class of Non-Uniformly Elliptic PDEs

TL;DR

This paper extends the theorem of sums to a broader class of non-uniformly elliptic PDEs by introducing a new compactness condition, enhancing the analytical tools available for viscosity solutions.

Contribution

It introduces a novel compactness condition that allows the theorem of sums to be applied to non-uniformly elliptic PDEs, expanding existing theoretical frameworks.

Findings

The compactness argument applies to a larger class of PDEs.

The theorem of sums can be extended beyond uniformly elliptic cases.

Provides new insights into viscosity solutions for degenerate elliptic PDEs.

Abstract

In the theory of viscosity solutions for second-order, degenerate elliptic PDEs, the Ishii-Lions method is a commonly used strategy, and the theorem of sums is the main analytical tool. As noted by Porretta and Priola, uniformly elliptic PDEs admit a version of the theorem of sums without the squared term due to a compactness argument. Here, we introduce a larger class of PDEs for which the compactness argument holds.