Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On Aviles-Giga limit states with $L^p$ entropy productions

Xavier Lamy, Andrew Lorent, Guanying Peng

Abstract

The Aviles-Giga energy provides sequences of maps converging to weak solutions $m\colonΩ\subset\mathbb R^2\to\mathbb R^2$ of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(Ω),\quad |m|=1\text{ a.e. in }Ω\,, \end{align*} whose entropy productions $\mathrm{div}\,Φ(m)$ are Radon measures in $Ω$, controlled by the energy. Here, the entropies are all $C^2$ vector fields $Φ\colon\mathbb S^1\to\mathbb R^2$ such that $\mathrm{div}\,Φ(m_*)=0$ for any smooth solution $m_*$. It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of $m$, as follows from the chain rule if $m$ has bounded variation. In particular, the entropy production measures should vanish if they coincide with $L^p$ functions: this is what we establish in this note if $p$ is not too small and under natural boundary conditions.

On Aviles-Giga limit states with $L^p$ entropy productions

Abstract

The Aviles-Giga energy provides sequences of maps converging to weak solutions of the eikonal equation \begin{align*} \mathrm{div}\, m=0\text{ in }\mathcal D'(Ω),\quad |m|=1\text{ a.e. in }Ω\,, \end{align*} whose entropy productions are Radon measures in , controlled by the energy. Here, the entropies are all vector fields such that for any smooth solution . It is conjectured that the entropy production measures are concentrated on the one-dimensional jump set of , as follows from the chain rule if has bounded variation. In particular, the entropy production measures should vanish if they coincide with functions: this is what we establish in this note if is not too small and under natural boundary conditions.

Paper Structure

This paper contains 8 sections, 6 theorems, 48 equations.

Table of Contents

  1. Introduction
  2. Acknowledgments.
  3. Proof of Theorem \ref{['t:besov_reg']}
  4. A kinetic formulation
  5. A coercive quantity
  6. A compensation identity
  7. Estimates for the right-hand side of the compensation identity
  8. Regularity estimate

Key Result

Theorem 1.2

Let $\Omega\subset{\mathbb R}^2$ be a bounded, $C^2$, simply connected open set, and $m\colon \Omega\to{\mathbb R}^2$ a weak solution of the eikonal equation eq:eikonal subject to the tangential condition eq:tangent_layer. $\blacktriangleleft$$\blacktriangleleft$

Theorems & Definitions (14)

  • Remark 1.1
  • Theorem 1.2
  • Theorem 1.3
  • proof : Proof of Theorem \ref{['t:Lpent_tangent']} from Theorem \ref{['t:besov_reg']}
  • Remark 1.4
  • Remark 2.1
  • Lemma 2.2
  • proof : Proof of Lemma \ref{['l:coercXivarphi']}
  • Lemma 2.3
  • proof : Proof of Lemma \ref{['l:comp_id']}
  • ...and 4 more