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

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

2007-03-23

arxiv.org/abs/math/0703689v1

Mathematics

Analysis of PDEs

25 pages

Scientific paper

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.

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