Stationary layered solutions for a system of Allen-Cahn type equations

TL;DR

This paper proves the existence of infinitely many layered solutions for a class of elliptic systems with symmetric double well potentials using variational methods, extending known results from one-dimensional to two-dimensional cases.

Contribution

It establishes the existence of infinitely many layered solutions for a class of elliptic systems based on the discrete structure of one-dimensional connecting solutions.

Findings

Existence of infinitely many layered solutions.

Application of variational methods to elliptic systems.

Extension from 1D to 2D solutions.

Abstract

We consider a class of semilinear elliptic system of the form $-\Delta u(x,y)+\nabla W(u(x,y))=0,\quad (x,y)\in\R^{2}$ where $W:\R^{2}\to\R$ is a double well non negative symmetric potential. We show, via variational methods, that if the set of solutions to the one dimensional system $-\ddot q(x)+\nabla W(q(x))=0,\ x\in\R$, which connect the two minima of $W$ as $x\to\pm\infty$ has a discrete structure, then the given system has infinitely many layered solutions.