Average gradient localisation for degenerate elliptic equations in the plane

Abstract

We consider Lipschitz solutions to the possibly highly degenerate elliptic equation $\mathrm{div}G(\nabla u)=0$ in $B_1\subset {\mathbb{R}}^2$, for any continuous strictly monotone vector field $G:{\mathbb{R}}^2\to {\mathbb{R}}^2$. We show that $u$ is either $C^1$ at $0$, or any blowup limit $v(x)=\lim \frac{u(\delta x)-u(0)}{\delta }$ along a sequence $\delta \to 0$ satisfies $\nabla v\in \mathcal{D}\cap \mathcal{S}\text{ a.e}$. Here, $\mathcal{D}$ and $\mathcal{S}$ can be roughly interpreted as the sets where ellipticity degenerates from below and above, that is, the symmetric parts of $\nabla G$ and $(\nabla G{)}^{-1}$ have a zero eigenvalue. This is a strong indication in favor of the expected continuity of $H(\nabla u)$ for any continuous $H$ vanishing on $\mathcal{D}\cap \mathcal{S}$. In contrast with previous results in the same spirit, we do not make any assumption on the structure of $G$ besides its continuity and strict monotony.