2007/03/23 by M. Röger, Röger, M., Y. Tonegawa +1
Mathematics · Physics and Astronomy · #35R35 #80A22 #Analysis of PDEs (math.AP) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Primary 49Q20 #Secondary 35B25 #math-ph #math.AP #math.MP #msc:35B25 #msc:35R35 #msc:49Q20 #msc:80A22
paper · pdf · doi:10.48550/arxiv.math/0703689
25 pages
arxiv created 2007/03/23 · arxiv updated 2009/12/01
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.