Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

On the $ Γ$-convergence of the Allen-Cahn functional with boundary conditions

Dimitrios Gazoulis

TL;DR

The paper establishes the $\Gamma$-convergence of the vector Allen-Cahn $\varepsilon$-functional with Dirichlet boundary data to a BV-partition energy $\tilde{J}_0$, incorporating a boundary trace term. It proves that, for limiting boundary data with three connected phases, the minimizer of the limiting energy is unique and the interfaces form a triod with $120^\circ$ angles in the interior of the disk; it also extends to mass-constraint settings. In the disk with equal interfacial tensions, the minimizer is explicitly the triod, and the energy scales as $3\sigma R$; in dimension three, the limiting minimizers are minimal cones (triod$\times\mathbb{R}$ or tetrahedral cones). The results connect the Allen-Cahn framework to geometric partition problems, provide precise structure results for minimizers under boundary data, and establish the intrinsic link between phase transitions and partition-based perimeter energies with boundary effects.

Abstract

We study minimizers of the Allen-Cahn system. We consider the $ \varepsilon $-energy functional with Dirichlet values and we establish the $ Γ$-limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves, we determine the precise structure of the minimizers of the limiting functional, and thus the limit of minimizers of the $ \varepsilon $-energy functional as $ \varepsilon \rightarrow 0 $.

On the $ Γ$-convergence of the Allen-Cahn functional with boundary conditions

TL;DR

The paper establishes the -convergence of the vector Allen-Cahn -functional with Dirichlet boundary data to a BV-partition energy , incorporating a boundary trace term. It proves that, for limiting boundary data with three connected phases, the minimizer of the limiting energy is unique and the interfaces form a triod with angles in the interior of the disk; it also extends to mass-constraint settings. In the disk with equal interfacial tensions, the minimizer is explicitly the triod, and the energy scales as ; in dimension three, the limiting minimizers are minimal cones (triod or tetrahedral cones). The results connect the Allen-Cahn framework to geometric partition problems, provide precise structure results for minimizers under boundary data, and establish the intrinsic link between phase transitions and partition-based perimeter energies with boundary effects.

Abstract

We study minimizers of the Allen-Cahn system. We consider the -energy functional with Dirichlet values and we establish the -limit. The minimizers of the limiting functional are closely related to minimizing partitions of the domain. Finally, utilizing that the triod and the straight line are the only minimal cones in the plane together with regularity results for minimal curves, we determine the precise structure of the minimizers of the limiting functional, and thus the limit of minimizers of the -energy functional as .
Paper Structure (12 sections, 15 theorems, 83 equations)

This paper contains 12 sections, 15 theorems, 83 equations.

Table of Contents

  1. Introduction
  2. Main Results
  3. Previous fundamental contributions
  4. Preliminaries
  5. Specialized definitions and theorems for the $\Gamma-$limit
  6. Specialized definitions and theorems for the Geometric problem
  7. Basic Lemmas
  8. Proof of the $\Gamma$-limit
  9. Minimizing partitions and the structure of the minimizer
  10. The structure of the minimizer in the disk
  11. Minimizers in dimension three
  12. Minimizers in the disc for the mass constraint case

Key Result

Theorem 1.1

Let $J_{\varepsilon}$ be defined by EnergyFunctional and $\tilde{J}_{\varepsilon} \;,\: \tilde{J}_0$ defined in EneryFunctionalwithBC and tildeJ_0 respectively. Then

Theorems & Definitions (32)

  • Theorem 1.1
  • Remark 1.2
  • Theorem 1.3
  • Lemma 2.1
  • Theorem 2.2
  • Remark 2.3
  • Definition 2.4
  • Proposition 2.5
  • Theorem 2.6
  • Lemma 3.1
  • ...and 22 more