Clifford Tori and the singularly perturbed Cahn–Hilliard equation

Abstract

In this paper we construct entire solutions u ε to the Cahn-Hilliard equation −ε 2 ∆(−ε 2 ∆u + W ′ (u)) + W ′′ (u)(−ε 2 ∆u + W ′ (u)) = 0, under the volume constraint R 3 (1 − u ε )dx = 4 √ 2π 2 , whose nodal set approaches the Clifford Torus, that is the Torus with radii of ratio 1/ √ 2 embedded in R 3 , as ε → 0. What is crucial is that the Clifford Torus is a Willmore hypersurface and it is non-degenerate, up to conformal transformations. The proof is based on the Lyapunov-Schmidt reduction and on careful geometric expansions of the laplacian.Keywords: Lyapunov-Schmidt reduction; Cahn-Hilliard equation; Willmore surface; Clifford Torus. IntroductionThe Allen-Cahn equationarises in several physical contexts, such as the study of the stable configurations of two different fluids confined in a bounded container Ω. If u(x) is the density of one of the two fluids at a point x ∈ Ω and the energy per unit volume is given by a function W of u, it looks reasonable to obtain stable configurations by minimizing the energy functionalamong all distributions fulfilling the volume constraintIf, for instance, W (u) = (1 − u 2 ) 2 , and m ∈ (−1, 1), any piecewise constant function taking only the values ±1 and satisfying (2) is a minimizer, irrespectively of the shape of the interface. Therefore this model is unsatisfactory, since it is very far from the reasonable physical assumption that the interfaces are area minimizers, so one replaces the energy byWe can see that there is a competition between the potential energy, that forces u to be close to ±1, and the gradient term that penalizes the phase transition. By minimizing this functional, we are looking for the physical interfaces in which the phase transition can occur. The minimizers u ε of E ε are solutions to the Euler Lagrange equation, that is (1). In order to see if the interfaces are actually minimal surfaces, it is interesting to study the asymptotic behaviour of the level sets {u ε = c} as the parameter ε → 0. It is useful to exploit the variational structure of the problem. It was shown by Modica and 2 Mortola that the energy E ε , seen as a functional on L 1 (Ω) and extended to be +∞ when the integrand is not an L 1 function, Γ−converges to the functional E(u) = cP er Ω ({u = 1}) if u = ±1 a.e. in Ω +∞ otherwise in L 1 (Ω) in the strong topology of L 1 (Ω) (see [23]), where c > 0 is a suitable constant. Moreover, Modica showed that, if u ε are minimizers of F ε under the volume constraintfor some m ∈ (−1, 1), then there exists a sequence ε k → 0 such that u ε k converges to some function u in L 1 (Ω) (see proposition 3 of [22]). Furthermore, Theorem 1 of [22] asserts that u = ±1 a. e. in Ω, and the set E = {x ∈ Ω : u(x) = 1} is actually a perimeter minimizer between all the subsets F ⊂ Ω satisfying the volume constraintFurther results about the relation between the minimizers of E ε and the minimizers of the perimeter can be found in [22] and in [7], where Choksi and Sternberg also described the relation between phase transition theory and the ...

Periodic Solutions to a Cahn–Hilliard–Willmore Equation in the Plane

“…provided the constant C 1 is large enough. This step of the proof is similar to that in Section 6, 2 of [35]. In order to prove existence, we have to show that S 1 maps the ball into itself, provided the constant is large enough, and that it is Lipschitz continuous in V with Lipschitz constant of order ε.…”

“…It is possible to find further details in [15], where the periodic case is specifically treated. This latter issue is common to interface constructions, see for example [34] and [35], and treated in a similar manner. Now we set φ := R 1/ε φ and we consider the change of variables t := z − φ (εs), which corresponds to replacing φ by φ in the expansion of the Laplacian (22).…”

“…As in [35], we want to use Fermi coordinates near a normal perturbation of the dilated periodic curve γ T . To this end, we fix φ ∈ C 4,ᾱ T (R) (recall (13)) such that φ C 4,α (R) < 1 , and for T large (i.e.…”

“…, where, we recall, φ and its derivatives are evaluated at εs. Hence by (20) the expansion we are interested in, using the latter coordinates (s, t), is given by (28) and (29) in [35].…”

“…Due to a recent result in [30], establishing the so-called Willmore conjecture, this torus minimizes the Willmore energy among all surfaces of positive genus. In [35], up to a small Lagrange multiplier, solutions of (3) in R 3 were found with interfaces approaching a Clifford torus, converging to −1 in its interior and to +1 on its exterior.…”