Layered solutions to the vector Allen-Cahn equation in $\mathbb{R}^2$. Minimizers and heteroclinic connections

Let $W:\mathbb{R}^m\to \mathbb{R}$ be a nonnegative potential with exactly two nondegenerate zeros $a_-≠ a_+∈\mathbb{R}^m$. We assume that there are $N≥q 1$ distinct heteroclinic orbits connecting $a_-$ to $a_+$ represented by maps $\bar{u}_1,...,\bar{u}_N$ that minimize the one-dimensional energy $J_\mathbb{R}(u)=∈t_\mathbb{R}(\frac{\vert u^\prime\vert^2}{2}+W(u)){d} s$.

We first consider the problem of characterizing the minimizers $u:\mathbb{R}^n\to \mathbb{R}^m$ of the energy $\mathcal{J}_Ω(u)=∈t_Ω(\frac{\vert\nabla u\vert^2}{2}+W(u)){d} x$. Under a nondegeneracy condition on $\bar{u}_j$, $j=1,...,N$ 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 $u:\mathbb{R}^2\to \mathbb{R}^m$ is necessarily an heteroclinic connection between suitable translates $\bar{u}_-(·-η_-)$ and $\bar{u}_+(·-η_+)$ of some $\bar{u}_±∈\{\bar{u}_1,...,\bar{u}_N\}$.

Then we focus on the existence problem and assuming $N=2$ and denoting $\bar{u}_-,\bar{u}_+$ the representations of the two orbits connecting $a_-$ to $a_+$ we give a new proof of the existence (first proved in [32]) of a solution $u:\mathbb{R}^2\to \mathbb{R}^m$ of

$Δ u=W_u(u),$

that connects certain translates of $\bar{u}_±$.