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

Abstract

Let $W:R^m\to R$ be a nonnegative potential with exactly two nondegenerate zeros $a_{-}\ne a_{+}\in R^m$. We assume that there are$N\ge 1$ distinct heteroclinic orbits connecting $a_{-}$ to $a_{+}$ represented by maps $u_1,\dots ,u_N$ that minimize the one-dimensional energy $J_R(u)={\int }_R(\frac{|u^{\prime }{|}^2}{2}+W(u))ds$. We first consider the problem of characterizing the minimizers $u:R^n\to R^m$ of the energy ${\mathcal{J}}_{\Omega }(u)={\int }_{\Omega }(\frac{|\nabla u{|}^2}{2}+W(u))dx$. Under a nondegeneracy condition on $u_1,\dots ,u_N$ 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 $u:R^n\to R^m$ is necessarily an heteroclinic connection between suitable translates $u_{-}(.-{\eta }_{-})$ and $u_{+}(.-{\eta }_{+})$ of some $u_{\pm }\in \{u_1,\dots ,u_N\}$. 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 $u:R^2\to R^m$ of \[Δu = W_u(u),\] that connects certain translates of $u_{\pm }$.