Existence of a solution to a vector-valued Allen-Cahn equation with a three well potential
arXiv:math/0702662
Abstract
In this paper we prove existence of a vector-valued solution 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 and the angles are determined by the function . This solution is an energy minimizer.
44 pages