On the Triple Junction Problem without Symmetry Hypotheses

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