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We have decided to discontinue the publication of preprints on our preprint server as of 1 March 2024. The publication culture within mathematics has changed so much due to the rise of repositories such as ArXiV (www.arxiv.org) that we are encouraging all institute members to make their preprints available there. An institute's repository in its previous form is, therefore, unnecessary. The preprints published to date will remain available here, but we will not add any new preprints here.

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