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Connectivity of the diffuse interface and fine structure of minimizers in the Allen-Cahn theory of phase transitions

Giorgio Fusco

Abstract

In the Allen-Cahn theory of phase transitions, minimizers partition the domain in subregions, the sets where a minimizer is near to one or to another of the zeros of the potential. These subregions that model the phases are separated by a tiny Diffuse Interface. Understanding the shape of this diffuse interface is an important step toward the description of the structure of minimizers. We assume Dirichlet data and present general conditions on the domain and on the boundary datum ensuring the connectivity of the diffuse interface. Then we restrict to the case of two dimensions and show that the phases can be separated, in a certain optimal way, by a connected network with a well defined structure. This network is contained in the diffuse interface and is a priori unknown. Under general assumption on the potential and on the Dirichlet datum, we show that, if we assume that the phase are connected, then we can obtain precise information on the shape of the network and in turn a detailed description of the fine structure of minimizers. In particular we can characterize the shape and the size of the various phases and also how they depend on the surface tensions.

Connectivity of the diffuse interface and fine structure of minimizers in the Allen-Cahn theory of phase transitions

Abstract

In the Allen-Cahn theory of phase transitions, minimizers partition the domain in subregions, the sets where a minimizer is near to one or to another of the zeros of the potential. These subregions that model the phases are separated by a tiny Diffuse Interface. Understanding the shape of this diffuse interface is an important step toward the description of the structure of minimizers. We assume Dirichlet data and present general conditions on the domain and on the boundary datum ensuring the connectivity of the diffuse interface. Then we restrict to the case of two dimensions and show that the phases can be separated, in a certain optimal way, by a connected network with a well defined structure. This network is contained in the diffuse interface and is a priori unknown. Under general assumption on the potential and on the Dirichlet datum, we show that, if we assume that the phase are connected, then we can obtain precise information on the shape of the network and in turn a detailed description of the fine structure of minimizers. In particular we can characterize the shape and the size of the various phases and also how they depend on the surface tensions.
Paper Structure (7 sections, 14 theorems, 153 equations, 17 figures)

This paper contains 7 sections, 14 theorems, 153 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Proof of Theorem \ref{['connect']}.
  3. Proof of Theorem \ref{['network']}.
  4. The existence of the optimal network.
  5. A theorem on the fine structure of $u^\epsilon$.
  6. Application of Theorem \ref{['fine']}.
  7. Appendix

Key Result

Theorem 1.1

There is $\delta_0>0$ such that if $h_1$ and $h_2$ hold and moreover Then, if $\{\omega_i^{\epsilon,\delta} \}_{i\in I}$ is a family of connected components of $\bar{\Omega}\setminus\mathscr{I}^{\epsilon,\delta}$, $\bar{\Omega}\setminus\{\omega_i^{\epsilon,\delta} \}_{i\in I}$ is a connected set. In particular $\mathscr{I}^{\epsilon,\delta}$ and $\bar{\Omega}\setminu

Figures (17)

  • Figure 1: The diffuse interface $\mathscr{I}^{\epsilon,\delta}$ and the phases $\Omega_a$, $a\in A$.
  • Figure 2: The idea of the proof of Theorem \ref{['connect']}.
  • Figure 3: The points $x_1^\pm=q_t^{-1}(\gamma(s_1^\pm)$ and the arc $\eta:[s_1^-,s_1^+]\rightarrow\partial\omega_{i_1}$.
  • Figure 4: Examples of cases I) and II).
  • Figure 5: The picture refers to the case $N=\tilde{N}=2$ where $n_b=0$ and $n_s=1$. The black line $\mathscr{G}$ denotes one of the infinite line in $\mathscr{I}$ that separate $\Omega_{a_1}$ and $\Omega_{a_2}$. The red line denotes $\hat{\mathscr{G}}$, the shortest such line in $\bar{\mathscr{I}}$. Note that $\Omega_{a_1}$ not connected together with the particular structure of $\Omega_{a_2}$ force $\hat{\mathscr{G}}$ to describe twice the same segment. This is one of the singularities that, a priori, can be expected for the minimizing network $\hat{\mathscr{G}}$.
  • ...and 12 more figures

Theorems & Definitions (25)

  • Theorem 1.1
  • Theorem 1.2
  • Lemma 2.1
  • proof
  • Proposition 4.1
  • proof
  • Corollary 4.2
  • proof
  • Theorem 5.1
  • Proposition 5.2
  • ...and 15 more