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MiS Preprint
31/2007

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

Matthias Röger and Yoshihiro Tonegawa

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.

Received:
Mar 23, 2007
Published:
Mar 23, 2007
MSC Codes:
49Q20, 35B25, 35R35, 80A22
Keywords:
phase transitions, Singular Perturbations, Gibbs-Thomson Law, Cahn-Hilliard Energy, Cahn-Hilliard nergy, Cahn-Hilliard Energy

Related publications

inJournal
2008 Journal Open Access
Matthias Röger and Yoshihiro Tonegawa

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

In: Calculus of variations and partial differential equations, 32 (2008) 1, pp. 111-136