Allen-Cahn Solutions with Triple Junction Structure at Infinity

We construct an entire solution $U:{\mathbb{R}}^2\to {\mathbb{R}}^2$ to the elliptic system $$\mathrm{\Delta }U={\mathrm{\nabla }}_{u}W(U),$$ where $W:{\mathbb{R}}^2\to [0,\mathrm{\infty })$ is a triple-well' potential. This solution is a local minimizer of the associated energy \[ \int \frac{1}{2}|\nabla U|^2+W(U)\,dx \] in the sense that $U$ minimizes the energy on any compact set among competitors agreeing with $U$ outside that set. Furthermore, we show that along subsequences, the blowdowns’ of $U$ given by $U_R(x):=U(Rx)$ approach a minimal triple junction as $R\to \mathrm{\infty }$. Previous results had assumed various levels of symmetry for the potential and had not established local minimality, but here we make no such symmetry assumptions.