A weak comparison principle for solutions of very degenerate elliptic equations

TL;DR

This paper establishes a comparison principle for weak solutions of a class of degenerate elliptic equations, where the ellipticity degenerates on a set depending on the gradient, advancing understanding of such equations.

Contribution

It introduces a comparison principle applicable to elliptic quasilinear equations with gradient-dependent degeneracy, a novel extension in the theory of degenerate elliptic PDEs.

Findings

Proves a comparison principle for solutions with gradient degeneracy.

Extends existing theory to equations with ellipticity degenerating on sets depending on the gradient.

Provides tools for analyzing solutions in highly degenerate elliptic equations.

Abstract

We prove a comparison principle for weak solutions of elliptic quasilinear equations in divergence form whose ellipticity constants degenerate at every point where $\nabla u \in K$, where $K\subset \mathbb{R}^N$ is a Borel set containing the origin.