Vector Minimizers in ℝ²

Abstract

Let $W:{{\mathbb R}}^m\rightarrow {{\mathbb R}}$ be a nonnegative potential with exactly two nondegenerate zeros $a^-\neq a^+\in {{\mathbb R}}^m$. Assume that there are N ≥ 1 distinct heteroclinic orbits connecting a ⁻ to a ⁺, represented by maps $\bar {u}_1,\ldots ,\bar {u}_N$ that minimize the one-dimensional energy $J_{{\mathbb R}}(u)=\int _{{\mathbb R}}(\frac {\vert u^\prime \vert ^2}{2}+W(u)){d} s$. Under a nondegeneracy condition on $\bar {u}_j$, j = 1, …, N and in two space dimensions we characterize the minimizers $u:{{\mathbb R}}^2\rightarrow {{\mathbb R}}^m$ of the energy ${J}_\varOmega (u)=\int _\varOmega (\frac {\vert \nabla u\vert ^2}{2}+W(u)){d} x$ that converge uniformly to a ^± as one of the coordinates converges to ±∞. We prove that a bounded minimizer $u:{{\mathbb R}}^2\rightarrow {{\mathbb R}}^m$ is necessarily an heteroclinic connection between suitable translates $\bar {u}_-(\cdot -\eta _-)$ and $\bar {u}_+(\cdot -\eta _+)$ of some $\bar {u}_\pm \in \{\bar {u}_1,\ldots ,\bar {u}_N\}$. Then, assuming N = 2 and denoting $\bar {u}_-,\bar {u}_+$ representatives of the two orbits connecting a ⁻ to a ⁺ we give a new proof of the existence (first proved in Schatzman [40]) of a solution $u:{{\mathbb R}}^2\rightarrow {{\mathbb R}}^m$ of Δu = W _u(u), that connects certain translates of $\bar {u}_\pm$.