article

Title: Existence of a solution to a vector-valued Allen-Cahn equation with a three well potential

Authors: Mariel Saez Trumper

Issue: Volume 58 (2009), Issue 1, 213-268

Abstract:

$In this paper we prove the existence of a vector-valued solution $v$ to \begin{gather*} -\Delta v + \frac{\nabla_vW(v)}{2} = 0, \\\lim_{r\to\infty}v(r\cos\theta,r\sin\theta) = c_i\quad\mbox {for} \theta\in(\theta_{i-1},\theta_i),\end{gather*} where $W:\mathbb{R}^2\to\mathbb{R}$ is a non-negative function that attains its minimum $0$ at $\{c_i\}_{i=1}^3$, and the angles $\theta_i$ are determined by the function $W$. This solution is an energy minimizer.$