

Preprint 31/2007
Convergence of phase-field approximations to the Gibbs-Thomson law
Matthias Röger and Yoshihiro Tonegawa
Contact the author: Please use for correspondence this email.
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
Download full preprint: PDF (311 kB), PS ziped (270 kB)
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.