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Local minimizers in $3$d of vector Allen-Cahn with a quadruple junction

Abhishek Adimurthi, Peter Sternberg

TL;DR

The paper establishes the existence of 3D vector Allen–Cahn local minimizers whose diffuse interfaces converge to a four-phase partition with a quadruple junction formed by a tetrahedral cone. It uses the $\Gamma$-convergence of a vector Modica–Mortola energy to a weighted perimeter functional $E_0$ and proves that a non-symmetric tetrahedral cone partition is an isolated $L^1$ local minimizer of $E_0$ on a carefully constructed domain $\Omega$ with a boundary configuration satisfying a good-normals condition. A boundary infiltration lemma and a calibration argument verify the local minimality, and a constructive domain perturbation of the sphere provides the geometric realization needed for the result. By Baldo–Kohn–PS theory, the isolated minimizer structure yields corresponding local minimizers of the diffuse energy $E_\varepsilon$ for small $\varepsilon$, which converge to the quadruple-junction partition as $\varepsilon\to0$. This work advances the desingularization program for minimal interfaces by demonstrating a stable, four-phase junction in 3D without symmetry assumptions on the wells.

Abstract

For $Ω$ a perturbation of the unit ball in $\mathbb{R}^3$, we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy. The sequence converges in $L^1$ to a partition of $Ω$ whose skeleton is given by a tetrahedral cone and thus contains a quadruple point. This is accomplished by proving that the partition is an isolated local minimizer of a weighted perimeter problem arising as the associated $Γ$-limit of the sequence of Allen-Cahn functionals.

Local minimizers in $3$d of vector Allen-Cahn with a quadruple junction

TL;DR

The paper establishes the existence of 3D vector Allen–Cahn local minimizers whose diffuse interfaces converge to a four-phase partition with a quadruple junction formed by a tetrahedral cone. It uses the -convergence of a vector Modica–Mortola energy to a weighted perimeter functional and proves that a non-symmetric tetrahedral cone partition is an isolated local minimizer of on a carefully constructed domain with a boundary configuration satisfying a good-normals condition. A boundary infiltration lemma and a calibration argument verify the local minimality, and a constructive domain perturbation of the sphere provides the geometric realization needed for the result. By Baldo–Kohn–PS theory, the isolated minimizer structure yields corresponding local minimizers of the diffuse energy for small , which converge to the quadruple-junction partition as . This work advances the desingularization program for minimal interfaces by demonstrating a stable, four-phase junction in 3D without symmetry assumptions on the wells.

Abstract

For a perturbation of the unit ball in , we establish the existence of a sequence of local minimizers for the vector Allen-Cahn energy. The sequence converges in to a partition of whose skeleton is given by a tetrahedral cone and thus contains a quadruple point. This is accomplished by proving that the partition is an isolated local minimizer of a weighted perimeter problem arising as the associated -limit of the sequence of Allen-Cahn functionals.

Paper Structure

This paper contains 7 sections, 5 theorems, 123 equations, 9 figures.

Table of Contents

  1. Introduction.
  2. Preliminaries.
  3. Partitions with sets of finite perimeter.
  4. Two helpful tools
  5. Description of the Domain.
  6. The main result.
  7. Construction of a Suitable Domain

Key Result

Theorem 1.1

There exists a domain $\Omega\subset\mathbb R^3$ such that for all $\varepsilon$ sufficiently small, the energy $E_\varepsilon$ possesses an $L^1$ local minimizer $u_\varepsilon$. As $\varepsilon\to 0$, one has that $u_\varepsilon\to u_0$ in $L^1(\Omega)$ with $u_0$ given by uzero.

Figures (9)

  • Figure 1: Tangent cones classified in Taylor when weights are equal.
  • Figure 2: A tetrahedral cone.
  • Figure 3: Partition of the unit ball by the tetrahedral cone. Each connected region here represents one of the sets $\tilde{S}_i$ restricted to the unit ball.
  • Figure 4: One of the components of the partition of the unit ball depicted in \ref{['Peter1']}.
  • Figure 5: The 'troughs' and 'valleys' on the surface of the deformed sphere.
  • ...and 4 more figures

Theorems & Definitions (10)

  • Theorem 1.1
  • Definition 2.1
  • Lemma 2.2
  • Proposition 2.3: Relative isoperimetric inequality
  • proof
  • Remark 2.4
  • Theorem 3.1
  • Theorem 3.2
  • proof
  • proof : Proof of \ref{['isolated_local_minimizer']}