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Hyperbolic regularization effects for degenerate elliptic equations

Xavier Lamy, Riccardo Tione

TL;DR

This work addresses the regularity of Lipschitz solutions to $\mathrm{div}(G(Du))=0$ in two dimensions under highly degenerate ellipticity. By recasting the PDE as a differential inclusion with a compact elliptic set $K$ and employing gradient localization, topological degree, and a nonlinear Beltrami–Hamilton–Jacobi framework, the authors establish a gradient localization theorem and show the singular set has $\mathcal{H}^1$-negligibility. They then analyze inclusions into elliptic curves and, when degeneracy concentrates on curves, obtain sharp partial $C^1$ regularity with a precise description of the singular set, using entropy productions and kinetic formulations. Finally, under structural assumptions on the degeneracy set (notably when components lie on curves or are simply connected/convex-boundary related), they derive broader partial regularity results and characterize blow-up limits, connecting to Monge-Ampère-type measures and Aviles–Giga-type theories, thereby extending prior zero-dimensional degeneracy results to curve-like degeneracies.

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.

Hyperbolic regularization effects for degenerate elliptic equations

TL;DR

This work addresses the regularity of Lipschitz solutions to in two dimensions under highly degenerate ellipticity. By recasting the PDE as a differential inclusion with a compact elliptic set and employing gradient localization, topological degree, and a nonlinear Beltrami–Hamilton–Jacobi framework, the authors establish a gradient localization theorem and show the singular set has -negligibility. They then analyze inclusions into elliptic curves and, when degeneracy concentrates on curves, obtain sharp partial regularity with a precise description of the singular set, using entropy productions and kinetic formulations. Finally, under structural assumptions on the degeneracy set (notably when components lie on curves or are simply connected/convex-boundary related), they derive broader partial regularity results and characterize blow-up limits, connecting to Monge-Ampère-type measures and Aviles–Giga-type theories, thereby extending prior zero-dimensional degeneracy results to curve-like degeneracies.

Abstract

This paper investigates the regularity of Lipschitz solutions to the general two-dimensional equation with highly degenerate ellipticity. Just assuming strict monotonicity of the field 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 is -negligible. As a consequence, we derive new sharp partial regularity results under the assumption that 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.
Paper Structure (31 sections, 48 theorems, 178 equations)

This paper contains 31 sections, 48 theorems, 178 equations.

Table of Contents

  1. Introduction
  2. The differential inclusions point of view
  3. General results
  4. Regularity for inclusions in elliptic curves
  5. Regularity for solutions to degenerate equations
  6. Preliminaries
  7. General notation and some elementary facts
  8. Essential range and generalized differentials
  9. Three equivalent viewpoints
  10. Theorem \ref{['tint:1']}: elliptic Curves and separation
  11. Setup and reduction
  12. Preliminary results
  13. Topological degree
  14. Quasiregular mappings
  15. Approximation of the problem
  16. ...and 16 more sections

Key Result

Theorem 1.1

If $K \subset \mathbb R^{2\times 2}$ is a compact set fulfilling e:diffinc, then $K$ is elliptic.

Theorems & Definitions (94)

  • Definition
  • Remark
  • Theorem 1.1
  • Theorem 1.2
  • Theorem 1.3: KS08
  • Remark 1
  • Lemma : sverak93
  • Theorem A
  • Definition 1
  • Theorem B
  • ...and 84 more