Layered solutions to the vector Allen-Cahn equation in $ R^2$. Characterization of minimizers and a new approach to heteroclinic connections

TL;DR

This paper characterizes minimizers of the vector Allen-Cahn equation in two dimensions, providing a new approach to heteroclinic connections and proving the existence of solutions connecting specific heteroclinic orbits.

Contribution

It offers a novel characterization of minimizers under nondegeneracy conditions and introduces a new proof for the existence of heteroclinic solutions in the vector Allen-Cahn equation.

Findings

Bounded minimizers are heteroclinic connections between translates of heteroclinic orbits.

A new proof of existence for solutions connecting heteroclinic orbits is provided.

The approach applies in two-dimensional space under specific nondegeneracy conditions.

Abstract

Let $W:R^m\rightarrow R$ be a nonnegative potential with exactly two nondegenerate zeros $a_-\neq a_+\in R^m$. We assume that there are$ N\geq 1$ distinct heteroclinic orbits connecting $a_-$ to $a_+$ represented by maps $ u_1,\ldots,u_N$ that minimize the one-dimensional energy $J_R(u) =\int_R(\frac{\vert u^\prime\vert^2}{2}+W(u))ds$. We first consider the problem of characterizing the minimizers $u:R^n\rightarrow R^m$ of the energy $\mathcal{J}_\Omega(u) =\int_\Omega(\frac{\vert\nabla u\vert^2}{2}+W(u))dx$. Under a nondegeneracy condition on $ u_1,\ldots,u_N $ and in two space dimensions, we prove that, provided it remains away from $a_-$ and $a_+$ in corresponding half spaces $S_-$ and $S_+$, a bounded minimizer $u:R^n\rightarrow R^m$ is necessarily an heteroclinic connection between suitable translates $ u_-(. -\eta_-)$ and $ u_+(. -\eta_+) $ of some $ u_\pm\in\{ u_1,\ldots, u_N\}$. Then we focus on the existence problem and assuming $N = 2$ and denoting $ u_-$ and $ u_+$ the representations of the two orbits connecting $ a_-$ to $ a_+$ we give a new proof of the existence (first proved in [31]) of a solution $ u:R^2\rightarrow R^m $ of \[\Delta u = W_u(u),\] that connects certain translates of $ u_\pm $.