On the Triple Junction Problem without Symmetry Hypotheses

We investigate the Allen–Cahn system \(\Delta u-W_u(u)=0\) $\mathrm{\Delta }u-W_u(u)=0$ , \(u:\mathbb {R}^2\rightarrow \mathbb {R}^2\) $u:{\mathbb{R}}^2\to {\mathbb{R}}^2$ , where \(W\in C^2(\mathbb {R}^2,[0,+\infty ))\) $W\in C^2({\mathbb{R}}^2,[0,+\infty ))$ 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 \(u_\varepsilon \) $u_{\epsilon }$ of the variational problem \(\min \int _{B_1} \left( \frac{\varepsilon }{2}\vert \nabla u\vert ^2+\frac{1}{\varepsilon }W(u) \right) \,\textrm{d}z\) $min{\int }_{B_1}\left(\frac{\epsilon }{2},{|\mathrm{\nabla }u|}^2,+,\frac{1}{\epsilon },W,(u)\right)\text{d}z$ , \(u=g_\varepsilon \) $u=g_{\epsilon }$ on \(\partial B_1\) $\partial B_1$ for some suitable boundary data \(g_\varepsilon \) $g_{\epsilon }$ . 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 or on the potential.