On some elementary properties of vector minimizers of the Allen-Cahn energy

Giorgio Fusco

doi:10.3934/cpaa.2014.13.1045

Article Contents

2014, Volume 13, Issue 3: 1045-1060. Doi: 10.3934/cpaa.2014.13.1045

This issue Previous Article Homogenisation theory for Friedrichs systems Next Article Multi-valued solutions to a class of parabolic Monge-Ampère equations

On some elementary properties of vector minimizers of the Allen-Cahn energy

Giorgio Fusco^1,

1.
Dipartimento di Matematica Pura e Applicata, Università de L’Aquila, I-67100 L’Aquila

Received: February 28, 2013

Revised: August 31, 2013

Published: April 2015

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Abstract

We derive a point-wise estimate for a map $u: \Omega \subset R^n \rightarrow R^m$ that minimizes $J_A(v): \int_A \frac{1}{2}|\nabla v|^2+U(v)$ subjected to the Dirichlet condition $v=u$ on $\partial\Omega$ for every open smooth and bounded set $A \subset \Omega$. We discuss some consequences of this basic estimate.

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Mathematics Subject Classification: Primary: 35J47, 35J50; Secondary: 35J20.

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References

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