Existence of a Solution to a Vector-valued Allen-Cahn Equation with a Three Well Potential

In this paper we prove the existence of a vector-valued solution v to $\begin{array}{c}-\mathrm{\Delta }\mathrm{v}+\frac{{\nabla }_{\mathrm{v}}\mathrm{W}\left(\mathrm{v}\right)}{2}=0,\\ \underset{\mathrm{r}\to \mathrm{\infty }}{\mathrm{lim}} \ \mathrm{v}(\mathrm{r} \ \mathrm{cos}\mathrm{\theta },\mathrm{r} \ \mathrm{sin}\mathrm{\theta })={\mathrm{c}}_{\mathrm{i}} \quad \mathrm{f}\mathrm{o}\mathrm{r} \ \mathrm{\theta }\in ({\mathrm{\theta }}_{\mathrm{i}-1},{\mathrm{\theta }}_{\mathrm{i}})\end{array}$, where W : ℝ2 → ℝ is a non-negative function that attains its minimum 0 at ${\left\{{\mathrm{c}}_{\mathrm{i}}\right\}}_{\mathrm{i}=1}^{3}$, and the angles θi are determined by the function W. This solution is an energy minimizer.