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 },\ma...

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Bibliographic Details
Published in:Indiana University mathematics journal 2009-01, Vol.58 (1), p.213-267
Main Author: Trumper, Mariel Saez
Format: Article
Language:English
Subjects:
Boundary conditions
Continuous functions
Critical points
Exact sciences and technology
General mathematics
General, history and biography
Grain boundaries
Infinity
Mathematical functions
Mathematical minima
Mathematics
Perceptron convergence procedure
Sciences and techniques of general use
Unit ball
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Summary: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.
ISSN:0022-2518
1943-5258
DOI:10.1512/iumj.2009.58.3233