Weak comparison principle for widely degenerate elliptic equations

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 } Ω,\notag \end{equation} where $p\ge 2$ and $\hat{I}\text{c◯}$ is an open subset of ${\mathbb{R}}^n$, $n\ge 2$. Moreover, we establish some second order regularity results of the solutions, that yields a weighted Sobolev inequality with widely degenerate weights.