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

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-CahnHilliard 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.

On the global minimizers of a nonlocal isoperimetric problem in two dimensions

“…REMARK 2.3 In Section 7.2 of [28], the authors establish C 3,α regularity of the reduced boundary of critical points of E γ that arise in the limit ε → 0 of critical points of the Ohta-Kawasaki energy E ε,γ given in (1.3).…”

The Willmore functional and instabilities in the Cahn-Hilliard equation

“…We will encounter this observation again in our numerical examples for small values of . For additional considerations of Γ− limits of this type see [31] and especially [11]. In the following we use the Willmore functional to detect local instabilities in finite time of transition solutions of (1.1) for small values of << 1.…”

Geometrization program of semilinear elliptic equations

“…In fact, it is known that for a family u ε of local minimizers of u ε with uniformly bounded energy must converge, up to subsequences, in L 1 -sense to a function of the form χ E − χ E c where χ denotes characteristic function, and ∂E has minimal perimeter. Thus the interface between the stable phases u = 1 and u = −1, represented by the sets [u ε = λ] with |λ| < 1 approach a minimal hypersurface, see Caffarelli and Córdoba [12,13], Hutchinson and Tonegawa [37], Röger and Tonegawa [65] for stronger convergence and uniform regularity results on these level surfaces.…”