On $C^1$ regularity for degenerate elliptic equations in the plane

TL;DR

This paper proves that Lipschitz solutions to certain degenerate elliptic equations in the plane are actually continuously differentiable, under mild conditions on the degeneracy set of the ellipticity, extending previous results.

Contribution

It extends regularity results for degenerate elliptic equations by allowing a finite degeneracy set where ellipticity degenerates both from below and above, using a novel transfer of estimates via a conjugate equation.

Findings

Lipschitz solutions are $C^1$ under mild degeneracy conditions.

The degeneracy set where ellipticity fails is finite.

A new method transfers estimates between different ellipticity regimes.

Abstract

We show that Lipschitz solutions $u$ of $\mathrm{div}\, G(\nabla u)=0$ in $B_1\subset\mathbb R^2$ are $C^1$, for strictly monotone vector fields $G\in C^0(\mathbb R^2;\mathbb R^2)$ satisfying a mild ellipticity condition. If $G=\nabla F$ for a strictly convex function $F$, and $0\leq \lambda(\xi)\leq \Lambda(\xi)$ are the two eigenvalues of $\nabla^2 F(\xi)$, our assumption is that the set $\lbrace\lambda=0\rbrace \cap \lbrace \Lambda=\infty\rbrace$, where ellipticity degenerates $both$ from below and from above, is finite. This extends results by De Silva and Savin (Duke Math. J. 151, No. 3, p.487-532, 2010), which assumed either that set empty, or the larger set $\lbrace \lambda=0\rbrace$ finite. Our main new input is to transfer estimates in $\lbrace \lambda > 0 \rbrace $ to estimates in $\lbrace \Lambda <\infty\rbrace$ by means of a conjugate equation. When $G$ is not a gradient, the ellipticity assumption needs to be interpreted in a specific way, and we highlight the nontrivial effect of the antisymmetric part of $\nabla G$.