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On the triple junction problem on the plane without symmetry hypotheses

Nicholas D. Alikakos, Zhiyuan Geng

Abstract

We investigate the Allen-Cahn system \begin{equation*} Δu-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where $W\in C^2(\mathbb{R}^2,[0,+\infty))$ is a potential with three global minima. We establish the existence of an entire solution $u$ which possesses a triple junction structure. The main strategy is to study the global minimizer $u_\varepsilon$ of the variational problem \begin{equation*} \min\int_{B_1} \left( \frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u) \right)\,dz,\ \ u=g_\varepsilon \text{ on }\partial B_1. \end{equation*} The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution.

On the triple junction problem on the plane without symmetry hypotheses

Abstract

We investigate the Allen-Cahn system \begin{equation*} Δu-W_u(u)=0,\quad u:\mathbb{R}^2\rightarrow\mathbb{R}^2, \end{equation*} where is a potential with three global minima. We establish the existence of an entire solution which possesses a triple junction structure. The main strategy is to study the global minimizer of the variational problem \begin{equation*} \min\int_{B_1} \left( \frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u) \right)\,dz,\ \ u=g_\varepsilon \text{ on }\partial B_1. \end{equation*} The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution.
Paper Structure (7 sections, 13 theorems, 183 equations, 3 figures)

This paper contains 7 sections, 13 theorems, 183 equations, 3 figures.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Lower bound for $J_\varepsilon(u_\varepsilon)$
  4. Localization of the transition layer
  5. Width of the transition layer
  6. Proof of Theorem \ref{['main theorem']}
  7. Proof of Lemma \ref{['lemma:upper bound']}

Key Result

Theorem 1.2

Fix $\gamma< \min\{\gamma_0,\min\limits_{i,j\in\{1,2,3\}} \frac{1}{2} \vert a_i-a_j\vert, \sqrt{\frac{\sigma}{20C_W}}\}$, where $\gamma_0, C_W$ are constants defined later. There is a constant $C_0$ which depends on $\gamma,W$ such that under the hypotheses (H1), (H2), there exists an entire, bounde

Figures (3)

  • Figure 1: red curve: $a_1$, green curve: $a_2$, blue curve: $a_3$, shaded region: $\Omega_1$, non-shaded region: $\Omega_2$.
  • Figure 2: Defintion of $l_1^t$, $l_2^t$. For any $z$ in the red shaded region, $|u(z)-a_1|\leq \gamma$.
  • Figure 3: $\hat{z}_1,\hat{z}_2\in\hat{ }^1_{\varepsilon,\gamma}$ and satisfy $\mathrm{dist}(\hat{z}_1, ^3_{\varepsilon,\gamma})\geq \mathrm{dist}(\hat{z}_1, ^2_{\varepsilon,\gamma})$, $\mathrm{dist}(\hat{z}_2, ^2_{\varepsilon,\gamma})\geq \mathrm{dist}(\hat{z}_2, ^3_{\varepsilon,\gamma})$ respectively. The figure on the right shows the enlargement of one portion of the transition layer between $a_1,a_2$. In an $O(\varepsilon)$ neighborhood of $\xi(h_j)$ (the blue region), $\vert u(z)-a_3\vert \leq 2\gamma$.

Theorems & Definitions (25)

  • Definition 1.1
  • Theorem 1.2
  • Lemma 2.1: Lemma 2.1 in AF
  • Lemma 2.2: Lemma 2.3 in AF
  • Lemma 2.3
  • Lemma 2.4
  • Proposition 3.1
  • proof
  • Proposition 3.2: lower bound of order $\varepsilon^{\frac{1}{2}}$
  • proof
  • ...and 15 more