Vector Minimizers in ℝ 2

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 = Wu(u), that connects certain translates of \(\bar {u}_\pm \).