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

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 $\mathcal{J}_Ω(u) =\int_Ω(\frac{\vert\nabla u\vert^2}{2}+W(u))dx$. 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 $u_-(. -η_-)$ and $u_+(. -η_+)$ 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 $Δu = W_u(u),$ that connects certain translates of $u_\pm$.

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