Preprint 23/1999

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 Edinburgh Society / 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 tex2html_wrap_inline7 in the plane. As tex2html_wrap_inline9 this functional favors tex2html_wrap_inline11 and penalizes singularities where tex2html_wrap_inline13 concentrates. Our main result is a compactness theorem: if tex2html_wrap_inline15 is uniformly bounded then tex2html_wrap_inline17 is compact in tex2html_wrap_inline19. Thus, in the limit tex2html_wrap_inline9 tex2html_wrap_inline23 solves the eikonal equation tex2html_wrap_inline11 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.

18.07.2014, 01:40