Regularity for quasilinear equations on degenerate singular sets

TL;DR

This paper establishes a universal gradient continuity estimate for solutions to quasilinear equations at their critical singular set, revealing enhanced regularity properties even where traditional elliptic regularity fails.

Contribution

It introduces a new regularity result showing solutions are as regular as constant coefficient solutions along the singular set, improving understanding of degenerate or singular equations.

Findings

Solutions are asymptotically as regular as constant coefficient solutions along the critical set.

Gradient of solutions has a superior modulus of continuity compared to the coefficients.

Provides a new perspective on smoothness properties beyond standard elliptic regularity.

Abstract

We prove a new, universal gradient continuity estimate for solutions to quasilinear equations with varying coefficients at points on its critical singular set of degeneracy $S(u) := \{X : D u(X) = 0 \}$. Our main Theorem reveals that along $S(u)$, $u$ is asymptotically as regular as solutions to constant coefficient equations. In particular, along the critical set $S(u)$, $Du$ enjoys a modulus of continuity much superior than the, possibly low, continuity feature of the coefficients. Our main, leading result fosters a new understanding on smoothness properties of solutions to degenerate or singular equations, beyond typical elliptic regularity estimates, precisely where the diffusion attributes of the equation collapse.