Phase field method for mean curvature flow with boundary constraints

This paper is concerned with the numerical approximation of mean curvature flow t → Ω(t) satisfying an additional inclusion-exclusion constraint Ω1 ⊂ Ω(t) ⊂ Ω2. Classical phase field model to approximate these evolving interfaces consists in solving the AllenCahn equation with Dirichlet boundary conditions. In this work, we introduce a new phase field model, which can be viewed as an Allen Cahn equation with a penalized double well potential. We first justify this method by a Γ-convergence result and then show some numerical comparisons of these two different models. Introduction In the last decades, a lot of work has been devoted to the motion of interfaces, and particularly to motion by mean curvature. Applications concern image processing (denoising, segmentation), material sciences (motion of grain boundaries in alloys, crystal growth), biology (modeling of vesicles and blood cells), image denoising, image segmentation and motion of grain boundaries. Let us introduce the general setting of mean curvature flows. Let Ω(t) ⊂ Rd, 0 ≤ t ≤ T , denote the evolution by mean curvature of a smooth bounded domain Ω0 = Ω(0) : the outward normal velocity Vn at a point x ∈ ∂Ω(t) is given by Vn = κ, (1) where κ denotes the mean curvature at x, with the convention that κ is negative if the set is convex. We will consider only smooth motions, which are well-defined if T is sufficiently small [3]. Since singularities may develop in finite time, one may need to consider the evolution in the sense of viscosity solutions [4, 13]. The evolution of Ω(t) is closely related to the minimization of the following energy: J(Ω) = ∫ ∂Ω 1 dσ. Indeed, (1) can be viewed as a L2-gradient flow of this energy. 1 Following [16, 15], the functional J can be approximated by a Ginzburg–Landau functional : Jǫ(u) = ∫ R ( ǫ 2 |∇u| + 1 ǫ W (u) ) dx. where ǫ > 0 is a small parameter, and W is a double well potential with wells located at 0 and 1 (for example W (s) = 1 2s 2(1− s)2). Modica and Mortola [16, 15] have shown the Γ-convergence of Jǫ to cWJ in L 1(Rd) (see also [5]), where cW = ∫ 1 0 √ 2W (s)ds. (2) The corresponding Allen–Cahn equation [2], obtained as the L2-gradient flow of Jǫ, reads ∂u ∂t = ∆u− 1 ǫ2 W (u). (3) Existence, uniqueness, and a comparison principle have been established for this equation (see for example chapters 14 and 15 in [3]). To this equation, one usually associates the profile q = argmin { ∫ R ( 1 2 γ 2 +W (γ) ) ; γ ∈ H loc(R), γ(−∞) = 1, γ(+∞) = 0, γ(0) = 1 2 } (4) Remark 1. The profile q (when W is continuous) can also be obtained [1] as the global decreasing solution of the following Cauchy problem