Weak comparison principle for widely degenerate elliptic equations

TL;DR

This paper establishes a comparison principle and second order regularity results for a class of widely degenerate elliptic equations, advancing understanding of solutions' behavior under degeneracy conditions.

Contribution

It introduces a comparison principle for weak solutions of widely degenerate elliptic equations and derives new second order regularity results and weighted Sobolev inequalities.

Findings

Proved a comparison principle for weak solutions.

Established second order regularity results.

Derived a weighted Sobolev inequality for degenerate weights.

Abstract

We prove a comparison principle for local weak solutions to a class of widely degenerate elliptic equations of the form \begin{equation} -\text{div} \left( \left(|Du|-1 \right)^{p-1}_+\frac{Du}{|Du|} \right) = f(x,u) \qquad \text{ in } \Omega,\notag \end{equation} where $p \ge 2$ and $\Omega$ is an open subset of $\mathbb{R}^{n}$, $n\geq2$. Moreover, we establish some second order regularity results of the solutions, that yields a weighted Sobolev inequality with widely degenerate weights.