Hyperbolic regularization effects for degenerate elliptic equations

TL;DR

This paper studies the regularity of Lipschitz solutions to a class of degenerate elliptic equations in two dimensions, revealing new partial regularity results by exploiting hyperbolic structures along degeneracy curves.

Contribution

It introduces a novel approach using hyperbolic structures and differential inclusions to establish partial regularity for solutions with degeneracy along curves.

Findings

Singular set of nondifferentiability points is negligible in measure.

Established pointwise gradient localization theorem.

Extended regularity results to cases where degeneracy occurs on curves.

Abstract

This paper investigates the regularity of Lipschitz solutions $u$ to the general two-dimensional equation $\text{div}(G(Du))=0$ with highly degenerate ellipticity. Just assuming strict monotonicity of the field $G$ and heavily relying on the differential inclusions point of view, we establish a pointwise gradient localization theorem and we show that the singular set of nondifferentiability points of $u$ is $\mathcal{H}^1$-negligible. As a consequence, we derive new sharp partial $C^1$ regularity results under the assumption that $G$ is degenerate only on curves. This is done by exploiting the hyperbolic structure of the equation along these curves, where the loss of regularity is compensated using tools from the theories of Hamilton-Jacobi equations and scalar conservation laws. Our analysis recovers and extends all the previously known results, where the degeneracy set was required to be zero-dimensional.