CONVERGENCE OF PHASE–FIELD APPROXIMATIONS TO THE GIBBS–THOMSON LAW
Preprint Series # 840
ROGER, MATTHIAS and TONEGAWA, YOSHIHIRO CONVERGENCE OF PHASE–FIELD APPROXIMATIONS TO THE GIBBS–THOMSON LAW. (2007);
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.