Existence of a solution to a vector-valued Allen-Cahn equation with a three well potential
In this paper we prove existence of a vector-valued solution $v$ to -Δv +\frac{\nabla_v W(v)}{2}&=0, \lim_{r\to \infty}v(r \cosθ,r\sinθ)&= c_i \hbox{for} θ\in (θ_{i-1}, θ_i), where $W:\rr^2\to \rr$ is non-negative function that attains its minimum 0 at $\{c_i\}_{i=1}^3$ and the angles $θ_i$ are determined by the function $W$. This solution is an energy minimizer.
math.AP↗