On the triple junction problem on the plane without symmetry hypotheses

We investigate the Allen-Cahn system \begin{equation*} Δu-W_u(u)=0,\quad u:$\mathbb{R}$^2\rightarrow$\mathbb{R}$^2, \end{equation*} where $W\in C^2({\mathbb{R}}^2,[0,+\mathrm{\infty }))$ is a potential with three global minima. We establish the existence of an entire solution $u$ which possesses a triple junction structure. The main strategy is to study the global minimizer $u_{\epsilon }$ of the variational problem \begin{equation*} \min\int_{B_1} \left( \frac{\varepsilon}{2}|\nabla u|^2+\frac{1}{\varepsilon}W(u) \right),dz,\ \ u=g_\varepsilon $\text{ on }$\partial B_1. \end{equation*} The point of departure is an energy lower bound that plays a crucial role in estimating the location and size of the diffuse interface. We do not impose any symmetry hypothesis on the solution.