Average gradient localisation for degenerate elliptic equations in the plane
Thibault Lacombe
TL;DR
This work studies regularity for planar degenerate elliptic equations of the form $\operatorname{div}(G(\nabla u))=0$ with $G$ continuous and strictly monotone. By combining a robust approximation scheme, localisation lemmas, and a topological analysis of the ellipticity/degeneracy sets $\mathcal{D}$ and $\mathcal{S}$, the authors prove a sharp dichotomy: at any point either $u$ is $C^1$ or any blow-up's gradient lies in $\mathcal{D}\cap\mathcal{S}$ a.e., and $\operatorname{dist}(\nabla u, \mathcal{D}\cap\mathcal{S})$ is approximately continuous. The results remove previous structural assumptions on the bad set, are two-dimensional in nature with zero right-hand side, and build on and extend the localisation framework of DSS and LacLam. The findings provide a precise description of blow-up limits and a near-continuity property for the distance to the degenerate/singular set, advancing the understanding of regularity for highly degenerate elliptic equations in the plane.
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(δx)-u(0)}δ $ along a sequence $δ\to 0$ 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.