Average gradient localisation for degenerate elliptic equations in the plane

TL;DR

This paper investigates the regularity of solutions to degenerate elliptic equations in the plane, showing that solutions are either smooth at a point or have blowup limits with gradients constrained to specific degenerate sets.

Contribution

It establishes a dichotomy for Lipschitz solutions to degenerate elliptic equations without structural assumptions on the vector field G, revealing gradient behavior at degeneracy.

Findings

Solutions are either $C^1$ at the origin or have blowups with gradients in degenerate sets.

Gradient limits satisfy conditions related to ellipticity degeneracy sets $\\mathcal{D}$ and $\mathcal{S}$.

Supports the conjecture that $H(\nabla u)$ is continuous when $H$ vanishes on degeneracy sets.

Abstract

We consider Lipschitz solutions to the possibly highly degenerate elliptic equation $ {\rm div} G(\nabla u)=0$ in $B_1\subset\mathbb{R}^2 $, for any continuous strictly monotone vector field $G \colon \mathbb{R}^2 \to \mathbb{R}^2$. 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.