Existence of a solution to a vector-valued Allen-Cahn equation with a three well potential

Details Existence of a solution to a vector-valued Allen-Cahn equation with a three well potential Existence of a solution to a vector-valued Allen-Cahn equation with a three well potential

2007-02-22

arxiv.org/abs/math/0702662v2

Mathematics

Analysis of PDEs

44 pages

Scientific paper

In this paper we prove existence of a vector-valued solution $v$ to -\Delta v +\frac{\nabla_v W(v)}{2}&=0, \lim_{r\to \infty}v(r \cos\theta,r\sin\theta)&= c_i \hbox{for} \theta \in (\theta_{i-1}, \theta_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 $\theta_i$ are determined by the function $W$. This solution is an energy minimizer.

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