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Another regularizing property of the 2D eikonal equation

Xavier Lamy, Andrew Lorent, Guanying Peng

TL;DR

This work establishes a sharp regularizing effect for weak solutions of the 2D eikonal equation under Besov regularity. It proves that if $m$ satisfies $|m|=1$, $ abla\cdot m=0$, and $m\in B^{1/3}_{6,\infty}$, then $m$ is locally Lipschitz away from a locally finite singular set, addressing the borderline Besov regime. Under disk tangential boundary conditions, the same regularity result holds for $q>\frac{47+\sqrt{553}}{12}$, leading to a rigidity statement: the only zero-energy state is the standard vortex $m(x)=i x/|x|$ in the inner disk. The analysis weaves together entropy productions, kinetic formulations, and refined Besov estimates to link low-regularity Besov data to strong local regularity, contributing to the Aviles–Giga program and the understanding of zero-energy states in planar eikonal-type problems.

Abstract

A weak solution of the two-dimensional eikonal equation amounts to a vector field $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$ such that $|m|=1$ a.e. and $\mathrm{div}\,m=0$ in $\mathcal D'(Ω)$. It is known that, if $m$ has some low regularity, e.g., continuous or $W^{1/3,3}$, then $m$ is automatically more regular: locally Lipschitz outside a locally finite set. A long-standing conjecture by Aviles and Giga, if true, would imply the same regularizing effect under the Besov regularity assumption $m\in B^{1/3}_{p,\infty}$ for $p>3$. In this note we establish that regularizing effect in the borderline case $p=6$, above which the Besov regularity assumption implies continuity. If the domain is a disk and $m$ satisfies tangent boundary conditions, we also prove this for $p$ slightly below $6$.

Another regularizing property of the 2D eikonal equation

TL;DR

This work establishes a sharp regularizing effect for weak solutions of the 2D eikonal equation under Besov regularity. It proves that if satisfies , , and , then is locally Lipschitz away from a locally finite singular set, addressing the borderline Besov regime. Under disk tangential boundary conditions, the same regularity result holds for , leading to a rigidity statement: the only zero-energy state is the standard vortex in the inner disk. The analysis weaves together entropy productions, kinetic formulations, and refined Besov estimates to link low-regularity Besov data to strong local regularity, contributing to the Aviles–Giga program and the understanding of zero-energy states in planar eikonal-type problems.

Abstract

A weak solution of the two-dimensional eikonal equation amounts to a vector field such that a.e. and in . It is known that, if has some low regularity, e.g., continuous or , then is automatically more regular: locally Lipschitz outside a locally finite set. A long-standing conjecture by Aviles and Giga, if true, would imply the same regularizing effect under the Besov regularity assumption for . In this note we establish that regularizing effect in the borderline case , above which the Besov regularity assumption implies continuity. If the domain is a disk and satisfies tangent boundary conditions, we also prove this for slightly below .
Paper Structure (10 sections, 12 theorems, 106 equations, 1 figure)

This paper contains 10 sections, 12 theorems, 106 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Improved estimate under tangent boundary conditions in a disk
  3. Plan of the article.
  4. Notation.
  5. Acknowledgments.
  6. Proof of Proposition \ref{['p:rigid_B3pconvol']}
  7. Proof of Lemma \ref{['l:characepsBr']}
  8. Proof of Lemma \ref{['l:badconvolB6']}
  9. Refined Besov estimate
  10. Proof of Theorem \ref{['t:bc']}

Key Result

Theorem 1.1

Let $\Omega\subset \mathbb{R}^2$ be an open set, and $m\colon\Omega\to{\mathbb R}^2$ a weak solution of the eikonal equation eq:eik. If $m\in B^{1/3}_{6,\infty}(\Omega)$, then $m$ is locally Lipschitz outside a locally finite set.

Figures (1)

  • Figure 1: Estimating the horizontal width of the thin band $L_\zeta +B_{\kappa\varepsilon^\alpha}$.

Theorems & Definitions (25)

  • Theorem 1.1
  • Remark 1.2
  • Proposition 1.3
  • Lemma 1.4
  • proof : Proof of Theorem \ref{['T1']} from Proposition \ref{['p:rigid_B3pconvol']} and Lemma \ref{['l:badconvolB6']}
  • Theorem 1.5
  • Remark 1.6
  • Proposition 2.1: GL20,LP23facto
  • Lemma 2.2
  • proof : Proof of Proposition \ref{['p:rigid_B3pconvol']} from Lemma \ref{['l:epsrigid_convol']}
  • ...and 15 more