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

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 $0\le \lambda (\xi )\le \Lambda (\xi )$ are the two eigenvalues of ${\nabla }^{2}F(\xi )$, our assumption is that the set $\{\lambda =0\}\cap \{\Lambda =\infty \}$, 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 $\{\lambda =0\}$ finite. Our main new input is to transfer estimates in $\{\lambda >0\}$ to estimates in $\{\Lambda <\infty \}$ 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 $\nabla G$.