Convergence of phase-field approximations to the Gibbs-Thomson law

Matthias Röger and Yoshihiro Tonegawa

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Submission date: 23. Mar. 2007 (revised version: July 2007)
Pages: 27
published in: Calculus of variations and partial differential equations, 32 (2008) 1, p. 111-136 
DOI number (of the published article): 10.1007/s00526-007-0133-6
Bibtex
MSC-Numbers: 49Q20, 35B25, 35R35, 80A22
Keywords and phrases: phase transitions, Singular Perturbations, Gibbs-Thomson Law, Cahn-Hilliard Energy, Cahn-Hilliard nergy, Cahn-Hilliard Energy
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Abstract:
We prove the convergence of phase-field approximations of the Gibbs-Thomson law. This establishes a relation between the first variation of the Van-der-Waals-Cahn-Hilliard energy and the first variation of the area functional. We allow for folding of diffuse interfaces in the limit and the occurrence of higher-multiplicities of the limit energy measures. We show that the multiplicity does not affect the Gibbs-Thomson law and that the mean curvature vanishes where diffuse interfaces have collided. We apply our results to prove the convergence of stationary points of the Cahn-Hilliard equation to constant mean curvature surfaces and the convergence of stationary points of an energy functional that was proposed by Ohta-Kawasaki as a model for micro-phase separation in block-copolymers.