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Half-space minimizing solutions of a two dimensional Allen-Cahn system

Zhiyuan Geng

TL;DR

This work provides a complete geometric and asymptotic description of half-space minimizers for a two-dimensional vector-valued Allen-Cahn system with Dirichlet boundary data. By analyzing blow-down limits, the authors classify all possible tangent profiles at infinity as constant, two-phase, or triple-junction partitions, with precise angular constraints governed by energy-minimizing Young-type laws. They prove the blow-down limit is unique and establish sharp approximations of the diffuse interface by the corresponding sharp interfaces, including convergence to one-dimensional heteroclinic connections along slices orthogonal to interfaces and exponential decay away from interfaces. The results connect diffuse-interface minimizers to minimal partition theory in the half-space and provide a detailed asymptotic picture of the interface geometry and profile near the boundary and interior interfaces.

Abstract

This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} Δu-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0 \text{ on } \partial \mathbb{R}_+^2, \end{equation*} where $W: \mathbb{R}^2\to [0,\infty)$ is a multi-well potential. We give a complete classification of such half-space minimizing solutions in terms of their blow-down limits at infinity. In addition, we characterize the asymptotic behavior of solutions near the associated sharp interfaces.

Half-space minimizing solutions of a two dimensional Allen-Cahn system

TL;DR

This work provides a complete geometric and asymptotic description of half-space minimizers for a two-dimensional vector-valued Allen-Cahn system with Dirichlet boundary data. By analyzing blow-down limits, the authors classify all possible tangent profiles at infinity as constant, two-phase, or triple-junction partitions, with precise angular constraints governed by energy-minimizing Young-type laws. They prove the blow-down limit is unique and establish sharp approximations of the diffuse interface by the corresponding sharp interfaces, including convergence to one-dimensional heteroclinic connections along slices orthogonal to interfaces and exponential decay away from interfaces. The results connect diffuse-interface minimizers to minimal partition theory in the half-space and provide a detailed asymptotic picture of the interface geometry and profile near the boundary and interior interfaces.

Abstract

This paper studies minimizing solutions to a two dimensional Allen-Cahn system on the upper half plane, subject to Dirichlet boundary conditions, \begin{equation*} Δu-\nabla_u W(u)=0, \quad u: \mathbb{R}_+^2\to \mathbb{R}^2,\ u=u_0 \text{ on } \partial \mathbb{R}_+^2, \end{equation*} where is a multi-well potential. We give a complete classification of such half-space minimizing solutions in terms of their blow-down limits at infinity. In addition, we characterize the asymptotic behavior of solutions near the associated sharp interfaces.
Paper Structure (11 sections, 25 theorems, 253 equations, 5 figures)

This paper contains 11 sections, 25 theorems, 253 equations, 5 figures.

Table of Contents

  1. Introduction
  2. Mathematical formulation and main results.
  3. Related results and outline of the paper
  4. Preliminaries
  5. Notations
  6. Basic estimates for vector-valued Allen-Cahn minimizers
  7. Classification of blow-down limits
  8. Uniqueness of the blow-down limit
  9. Asymptotic behaviors near the sharp interface
  10. Proof of the upper bound in Lemma \ref{['lem:sharp ene bd']}.
  11. Proof of Proposition \ref{['prop: existence of h']}

Key Result

Theorem 1.3

Let $u:\mathbb{R}_+^2\to \mathbb{R}^2$ be a minimizing solution of eq:2D allen cahn satisfying bdy cond--uniform bound. There exists a minimizing partition map $\tilde{u}\in \mathrm{BV}_{loc}(\mathbb{R}_+^2;\{a_1,a_2,...,a_N\})$ in the sense of Definition def: min par map such that $\nabla \tilde{u} where $u_r(z):=u(rz)$. Extend the definition of $\tilde{u}$ to the boundary $\partial \mathbb{R}_+^

Figures (5)

  • Figure 1: $\Omega_i$ for $i=1,...,5$
  • Figure 2: Case I: move the junction inside to reduce energy.
  • Figure 3: Case 4: move the junction along the angle bisector of $C_1$
  • Figure 4: Definition of $_0$.
  • Figure 5: Red: $a_1$, Green: $a_2$, Blue: $a_3$

Theorems & Definitions (50)

  • Definition 1.1
  • Definition 1.2
  • Theorem 1.3
  • Remark 1.1
  • Theorem 1.4
  • Remark 1.2
  • Lemma 2.1: Lemma 2.3 in AF
  • Lemma 2.2: Variational maximum principle, AF2
  • Lemma 2.3: AF3
  • Lemma 2.4
  • ...and 40 more