A compactness result in the gradient theory of phase transitions

Antonio DeSimone, Robert V. Kohn, Stefan Müller, and Felix Otto

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Submission date: 03. May. 1999
Pages: 16
published in: Proceedings of the Royal Society of Edinburgh / A, 131 (2001) 4, p. 833-844 
DOI number (of the published article): 10.1017/S030821050000113X
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Abstract:
We examine the singularly perturbed variational problem in the plane. As this functional favors and penalizes singularities where concentrates. Our main result is a compactness theorem: if is uniformly bounded then is compact in . Thus, in the limit solves the eikonal equation almost everywhere. Our analysis uses ``entropy relations'' and the ``div-curl lemma,'' adopting Tartar's approach to the interaction of linear differential equations and nonlinear algebraic relations.