On the triple junction problem on the plane without symmetry hypotheses

TL;DR

This paper proves the existence of a triple junction solution to the Allen-Cahn system with a potential having three minima, using variational methods and energy estimates without symmetry assumptions.

Contribution

It establishes the existence of a triple junction solution to the Allen-Cahn system without symmetry constraints, advancing understanding of phase transition interfaces.

Findings

Existence of a triple junction solution for the Allen-Cahn system.

Development of energy lower bounds for diffuse interface analysis.

Application of variational methods to non-symmetric solutions.

Abstract

We investigate the Allen-Cahn system \begin{equation*} \Delta u-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.