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

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


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