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Minimizers for boundary reactions: renormalized energy, location of singularities, and applications

Xavier Cabre, Neus Consul, Matthias Kurzke

Abstract

The Casten-Holland and Matano theorem for interior reactions states that no nonconstant stable solutions exist in convex domains $Ω$ of $\mathbb{R}^n$ under zero Neumann boundary conditions. In this paper we establish that the analogous statement fails for boundary reactions when $n=2$ (that is, for harmonic functions in $Ω$ with a Neumann reaction term on its boundary $\partialΩ$). For instance, nonconstant stable solutions exist when $Ω$ is a square, or a smooth strictly convex approximation of it. In regular polygons of many sides, which approach the circle, we can prove the existence of as many nonconstant stable solutions as wished. Instead, in the circle such stable solutions do not exist. More importantly, we can predict the existence or not of nonconstant stable solutions, as well as the location of its boundary "vortices" $(p,q)$, through the properties of a real function defined on $\partialΩ\times\partialΩ$ (the renormalized energy) which depends only on the conformal structure of the domain $Ω$. This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line.

Minimizers for boundary reactions: renormalized energy, location of singularities, and applications

Abstract

The Casten-Holland and Matano theorem for interior reactions states that no nonconstant stable solutions exist in convex domains of under zero Neumann boundary conditions. In this paper we establish that the analogous statement fails for boundary reactions when (that is, for harmonic functions in with a Neumann reaction term on its boundary ). For instance, nonconstant stable solutions exist when is a square, or a smooth strictly convex approximation of it. In regular polygons of many sides, which approach the circle, we can prove the existence of as many nonconstant stable solutions as wished. Instead, in the circle such stable solutions do not exist. More importantly, we can predict the existence or not of nonconstant stable solutions, as well as the location of its boundary "vortices" , through the properties of a real function defined on (the renormalized energy) which depends only on the conformal structure of the domain . This requires the development of a new Ginzburg-Landau theory for real-valued functions and the analysis of the half-Laplacian on the real line.
Paper Structure (12 sections, 43 theorems, 257 equations)

This paper contains 12 sections, 43 theorems, 257 equations.

Table of Contents

  1. Introduction
  2. Interior reactions under zero Neumann boundary conditions
  3. Boundary reaction problems
  4. Proof of the main theorems
  5. Plan of the paper
  6. The layer solution in the half-plane. Nonexistence of homoclinic solutions
  7. Renormalized energy for boundary values in ${\mathbb S}^0$
  8. Minimization in a closed neighborhood and energy upper bounds
  9. Pohoz̆aev balls and covering arguments
  10. Blow-up and the sharp lower bound
  11. Domains where the renormalized energy has nontrivial local minimizers
  12. Computing the value of $C_f$ in a special case

Key Result

Theorem 1.1

Let $\Omega\subset{\mathbb R}^n$ be a bounded, smooth, convex domain, and let $f\in C^1({\mathbb R})$. Assume $u\in C^2(\overline\Omega)$ is a solution of the Neumann problem and that $u$ is stable, in the sense that Then, $u$ is constant.

Theorems & Definitions (79)

  • Theorem 1.1: Casten:1978aaMatano:1979aa
  • Theorem 1.2
  • Theorem 1.3
  • Theorem 1.4
  • Theorem 1.5
  • Theorem 1.6
  • Proposition 1.7: proved later as Proposition \ref{['prop:ub']}
  • Proposition 1.8: proved later as Proposition \ref{['pro:conmin']}
  • Proposition 1.9: see Section \ref{['sec:poho']}, especially Proposition \ref{['pro:57']}
  • Proposition 1.10: proved as Proposition \ref{['pro:just2']}
  • ...and 69 more