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

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.


Introduction
Phase separation is a common phenomenon in many areas of the sciences. Alloys studied in material sciences, melting and solidification processes, or blockcopolymers investigated in physical chemistry, they all show the coexistence of two or more phases, separated by thin transition layers. The main approaches to describe phase transitions are on the one hand sharp interface models and on the other hand diffuse interface models, also referred to as 'phase field' or 'Ginzburg-Landau' models. The relation between both kinds of models remains an outstanding question. Rigorous passages to the sharp interface limit are often difficult and generalized formulations for the limit problems are necessary to obtain the convergence of diffuse approximations. However, care has to be taken that solutions satisfy the equations in a reasonably strong sense.
The goal of the present paper is to prove the convergence of diffuse approximations of the so-called Gibbs-Thomson law, which states that the mean curvature of the phase boundary is given as the trace of a function in the bulk. Our result relates the first variation of the Van-der-Waal-Cahn-Hilliard energy, which is the common root of most phase field models, to the first variation of the area functional. To the best of our knowledge, we give the first satisfactory solution in the case that diffuse interfaces collapse or cancel each other in the limit.
Before stating the main result we describe the setting and background of the problem.
1.1. Phase fields, sharp interfaces, and the Gibbs-Thomson law. The diffuse interface approach is based on a free energy that acts on smooth phase fields and that was proposed by Van-der-Waals [33] and later Cahn-Hilliard [7]. In a normalized form this energy is given by where ε > 0 is a small parameter and W is a nonnegative 'double-well potential' with value zero if and only if u = ±1. Domains where u ≈ 1 or u ≈ −1 represent two coexisting phases, separated by diffuse interfaces. Formal arguments show that E ε favors transition layers with a thickness of order ε. Hence, as ε tends to zero the diffuse interfaces become sharp. One naturally associated quantity to the Cahn-Hilliard energy is its L 2functional derivative, which often corresponds to the chemical potential, In many applications f ε is given by means of other quantities and a certain control on f ε is available. The corresponding functional derivative of the surface area functional, evaluated at a smooth compact hypersurface Σ, is given by the mean curvature of Σ and (1.2) formally corresponds to the following equation, in solidification processes known as Gibbs-Thomson law (and we will adopt this term throughout the paper), with a surface tension coefficient σ > 0. The Gibbs-Thomson law relates the local geometry of the phase boundary to a function f : Ω → R in the bulk, for example the temperature or the chemical potential.
1.2. Main results. Let us first state our main result in a concise form (we will prove a slightly stronger statement, given in Theorem 3.2).
Theorem 1.1. Suppose p > n and let sequences of functions {u ε } ε>0 ⊂ W 3,p (Ω) and functions {f ε } ε>0 ⊂ W 1,p (Ω) be given such that (1.2) holds and such that E ε (u ε ) ≤ Λ, (1.4) u ε → u in L 1 (Ω), (1.5) f ε → f weakly in Remark 1.2. For a sequence (u ε ) ε>0 that satisfies the uniform energy bound (1.4) and a sequence (f ε ) ε>0 that is uniformly bounded in W 1,p (Ω) there exists a subsequence ε → 0 such that (1.5), (1.6) hold. Besides these uniform bounds no other conditions, such as energy minimality, are required to apply Theorem 1.1. For this reason the result is relevant to a large class of stationary and timedependent problems. In section 7 we use our results to characterize the limit of stationary points of the Cahn-Hilliard functional (1.1) and to prove the convergence of stationary points in a model for block-copolymers.
The assumption (1.6) on the chemical potentials f ε is still restrictive. We conjecture that the (weak) convergence of f ε in W 1,p (Ω) with p > n/2 would suffice to conclude (1.7). However, our techniques yet require the continuity of f , which is ensured only if p > n. For the Cahn-Hilliard equation for example the natural regularity of the chemical potential is W 1,2 (Ω) in space. Hence, our result does not apply in this case.
1.3. Related results and main techniques. Since the fundamental work of Modica and Mortola [22,21] on the convergence of E ε to the area functional the relation between their first variations has drawn attention. Modica [21] and Sternberg [40] proved that minimizers of E ε under a volume constraint converge to area-minimizing hypersurfaces with an integral constraint. Luckhaus-Modica [19] then showed that the Lagrange-multipliers associated with the volume constraint converge to the constant mean curvature of the limiting hypersurface. Ilmanen [17] considered the corresponding L 2 -gradient flows and proved the convergence of the Allen-Cahn equation to the mean-curvature flow, in the varifold formulation of Brakke [5]. Convergence of various other phase field problems to the corresponding sharp interface models have been shown either formally or rigorously [8,1,27,6,18,9,38], sometimes in quite involved weak formulations.
The second author considered, partly in joint-work with Hutchinson, the convergence of diffuse interface approximations of the Gibbs-Thomson law, under different assumptions on the chemical potential [16,41,42]. However, the Gibbs-Thomson relation is only verified in an (in some respect unsatisfactory) multiplicitydependent formulation, see (1.9) and the discussion below.
Schätzle [35] considered a sequence of hypersurfaces with mean curvature given by a Sobolev function in the ambient space and obtained that the Gibbs-Thomson law holds in the limit in a rather clean varifold formulation. In [35] the chemical potentials f ε in (1.6) need only to converge in a Sobolev space f ∈ W 1,p (Ω) with p > n/2, c.f. Remark 1.2.
Geometric Measure Theory provides suitable generalized formulations in spaces that allow for the compactness of approximations. Luckhaus-Modica [19] and Luckhaus-Sturzenhecker [20] introduced a weak formulation of the Gibbs-Thomson law (1.3) for characteristic functions of bounded variation. This formulation is rather natural and has the advantage of being based directly on the phase function. However, justifying the Gibbs-Thomson law in the limit of approximations requires the additional assumption that no cancellation of (diffuse) interfaces occurs. Unfortunately, this property does in general not hold [34].
To master such cancellations Ilmanen [17] used a varifold-approach. He considered the limit of the diffuse surface-area measures (energy measures) The idea behind is that this limit makes information visible that is lost in the limit of the phase fields: Where cancellation of the approximate phase boundaries occurs the limit µ of the measures µ ε carries a higher multiplicity. The support of µ eventually extends the limit phase boundary by hidden boundaries. Showing that the limit measure is in fact given as a integer-rectifiable varifold with a weak mean curvature vector, the Gibbs-Thomson law can be verified in a varifold formulation. This strategy was used for various problems by Chen [9], Soner [39], Hutchinson-Tonegawa [16], Tonegawa [41,42], and others. However, in none of these papers the problem of higher multiplicity was completely solved. Typically the convergence of the diffuse phase fields and the diffuse surface-area measures µ ε is shown and the rectifiability of the limit µ as well as the existence of a weak mean curvature H µ is obtained. Still, the Gibbs-Thomson law holds only in a multiplicity-dependent formulation (1.9) where N is the density function of the measure µ. This formulation is for two reason unsatisfactory: First the Gibbs-Thomson law should be satisfied by the phase boundary rather than by the (in view of the applications) 'obscure' measure µ. Secondly, the density function N should not affect the Gibbs-Thomson law. To prove the full results (1.9) has to be complemented by In recent years progress has been made on this issue. Schätzle [35] proves the Gibbs-Thomson law in the limit of an approximation by hypersurfaces: There the weak mean curvature H µ of the limit measure µ satisfies (1.3) and (1.10). It was then shown by the first author [31] that H µ is in fact a property of the phase boundary ∂ * {u = 1}, see Appendix A. This is crucial in order to apply the (stationary) convergence result [35] to evolution problems [31,32]. The higher-multiplicity problem is even more challenging in the context of the sharp interface limit of diffuse approximations, due to the singular nature of this limit process. The three main ingredients of our proof are first an earlier result of the second author [42] on the convergence of certain phase field equations with chemical potential. This ensures rectifiability, existence of a weak mean curvature with appropriate regularity and the multiplicity-dependent Gibbs-Thomson relation (1.9). The second ingredient is the fine local analysis of Schätzle [36] on rectifiable measures with sufficiently regular weak mean curvature. The third important argument is a comparison principle for the phase fields u ε and diffuse approximations of suitably constructed comparison graphs.
1.4. Organization of the paper. In the next section we will precisely formulate our assumptions and introduce some notations. Section 3 states our main result. A localization step in Section 4 prepares a contradiction argument that we will use in Section 5 to prove our main Theorem 3.2. We first assume that a certain comparison principle, which is given in Proposition 6.11, holds. Section 6 is then devoted to the proof of this Proposition. Finally we give in Section 7 two applications of Theorem 1.1 and we recall in the appendix the definition of a generalized mean curvature for phase boundaries that we will use.

Notations and assumptions
We state first all assumptions and definitions, including those already appeared in the introduction.
Assumption 2.1. Consider a bounded domain Ω ⊂ R n with Lipschitz-boundary and the standard double-well potential W given by We define an energy functional E ε on W 1,2 (Ω), Assume further that We may generalize W to be any C 3 -function with two non-degenerate minima and one local maximum, so that the results in [42] apply.
We next associate diffuse surface-area measures and appropriate varifolds to the functions u ε . Definition 2.2. For u ε we define Radon-measures µ ε on Ω,
Finally we define the mean-curvature operator H for graphs: for p ∈ R n−1 , X ∈ S(n − 1) set
Our main results are summarized in the following theorem.

Localization
In this section we show that we can restrict ourselves to a 'generic' local situation, where the support of µ is well described in terms of graphs. We then apply a result of Schätzle [36] that gives a fine description of the varifold µ.
In addition we may assume without loss of generality that We fix x 0 such that (4.1)-(4.5) hold. After applying a suitable translation and rotation we may assume that x 0 = 0 and To apply a contradiction argument we assume (3.3), (3.4) to be false. By (2.8), (4.5) there exists 0 > 0 such that We distinguish four cases depending on whether f (0) < 0 or f (0) > 0 and whether u = 1 or u = −1 in the region 'above' supp(µ). In the following we consider the case that That implies that The other cases can be treated analogously or follow from a symmetry argument. By (2.9) we obtain that for all ε > 0 sufficiently small. In the next step we apply a result of Schätzle [36] on the local structure of the measure µ. First we need some definitions. Definition 4.3. We define the upper and lower height-functions ϕ + , ϕ − : with the convention that the supremum over an empty set is −∞ and the infimum over an empty set is +∞. Moreover we set and
We choose below a 'good point' for which we derive a contradiction to Assumption 4.2. Before, we need another definition.
Definition 4.5. We say that a function ψ has a second-order Taylor expansion at a point y 1 ∈ R n−1 if there exist p ∈ R n−1 , X ∈ S(n − 1) such that We then set ∇ψ(y 1 ) := p, D 2 ψ(y 1 ) := X.
The next Proposition is the heart of the contradiction argument. It relies on the fact that the approximations u ε behave as if the curvature of the limit interface is given by f /σ rather than by f /(N σ).
In the case that N 0 ≥ 1 this property can be proved in the same way as we will prove Proposition 5.2: One constructs smooth approximations v ε of the function 2X {t>ψ(y)} − 1 and uses a comparison principle to obtain u ε ≤ v ε . However, these arguments do not apply if N 0 ≥ 2 is even, since in that case u ε ≈ 1 is larger than 2X {t>ψ(y)} − 1 in the region 'below'ψ.
6. Proof of Proposition 5.2 Assume that (5.7) does not hold, that is
Since ψ is continuous and ϕ + is upper-semicontinuous there exist 0 As explained before we will use that u ε behaves as if the curvature of the sharp interface limit is given by f /σ, instead of f /(N σ). In a first step we construct functions v ε such that In the second step we will apply a comparison principle to u ε , v ε to obtain a contradiction in the limit ε → 0.
6.1. Construction of v ε . The two ingredients to construct v ε are a modified distance function from graph(ψ) and the optimal profile and first order-correction of the one-dimensional minimisation problem associated to the Cahn-Hilliard functional. Remark 6.2. Since ψ is smooth we deduce that M is a smooth hypersurface and that there exists δ > 0, δ = δ( ψ C 2 (B (y1)) , 2 ) such that the distance function d is unique and smooth in a neighborhood We turn to the optimal profile for the one-dimensional minimisation in the Cahn-Hilliard energy. and let φ 1 : R → R be the first order correction (see [26]), .
Since the distance function d is smooth only in a neighborhood of graph(ψ) we have to modify the distance function.
and such that the conditions are satisfied. Moreover we choose smooth functions β ε : R → R, ε > 0, with .
Proposition 6.11. For all ε > 0 sufficiently small we obtain that Proof. Let us define the sets Assume now ε < ε 1 , where we choose ε 1 > 0 below, and that (6.53) is not satisfied, hence The definition of v ε in (6.15), (6.23) implies that we can choose s 0 > 0, s 0 = s 0 ( 0 ) such that for all ε > 0 sufficiently small Applying then Lemma 6.9 with U, Ω as above and k = 2 we deduce that Therefore (6.51) and (6.55), (6.56) imply that and we deduce that for all ε < ε 1 and ε 1 > 0 chosen suitably small. Since Φ is continuous (6.54), (6.57) imply the existence of s * > 0 and x * ∈ U such that (6.58) We first prove that x * ∈ U . With this aim we consider the different parts of ∂U .
(1) For ε 1 > 0 chosen suitably small we have v (s * ) and therefore, by Lemma 6.9 and Lemma 6.10, for ε < ε 1 and ε 1 > 0 suitably small, see the argument above. This shows that (2) For ε 1 > 0 sufficiently small we obtain from Lemma 6.8 and Lemma 6.9, applied to By (6.50) this implies that By similar arguments as above we first prove that u ε > v (s * ) ε in U 1 . First we obtain from Lemma 6.8 that and applying Lemma 6.9 and Lemma 6.10 with k = 2 we deduce that in U 1 (6.65) Since s * > 0 we obtain that for ε < ε 1 , where in U 2 . By (6.64), (6.65), (6.66) we deduce that In the case that λ = 0 the phase boundary is up to a H n−1 -nullset a smooth hypersurface. The energy measures µ ε as defined in (2.6) converge to a measure µ that is up to the factor 2σ integer-rectifiable, has constant mean curvature σH = λ and multiplicity one H n−1 -almost everywhere on ∂ * {u = 1}. Moreover 'hidden boundaries' can only occur in one phase and have zero mean curvature: Proof. It follows from [9, Lemma 3.4] that |λ ε | ≤ c(m, Λ) and we may choose a subsequence such that λ = lim ε→0 λ ε exists. We therefore can apply Theorem 3.2 and obtain that there exists a subsequence ε → 0 and limits u, µ of u ε , µ ε . Moreover, u ∈ BV (Ω, {−1, 1}) and (2σ) −1 µ is an integer-rectifiable varifold with weak mean curvature σH = λ H n−1 − almost everywhere on ∂ * {u = 1}.
Next it follows from Theorem 3.2 that λ = 0 on the parts of ∂ * {u = 1}∩Ω with odd multiplicity larger than 1, which shows that in the case λ = 0 the phase boundary is given as a constant curvature varifold with unit multiplicity. By Allard's regularity theory [3] we conclude the smoothness of the phase boundary. Finally (7.5), (7.6) follow from (3.4).
In general dimension we can not insure good regularity of the hidden boundaries, due to the lack of regularity theory for general stationary integral varifold. Only for n = 2 we can conclude that spt µ ∩ K is given by straight line segments with possible junction points for all compact sets K ⊂ Ω \ ∂ * {u = 1} [2]. 7.2. Critical points of the Ohta-Kawasaki functional. The micro-phase separation of block copolymers exhibits the formation of complex patterns. Ohta-Kawasaki [25] and later Bahiana-Oono [4] used a phase-field like approach and proposed a free energy that is after a suitable rescaling given by where v(·) = v[u](·) is the solution of −∆v = u − 1 |Ω| Ω u in Ω, ∇v · ν Ω = 0 on ∂Ω. (7.8) The functional F ε extends the Cahn-Hilliard energy by a non-local term that describes long-range interactions between chains of macromolecules. For a derivation of this model by a density-functional approach see [11]. The set of (local) minimizers of F ε is extremely rich and (7.7) has drawn quite some attention [10,24,29,30]. The Gamma-limit of F ε as ε → 0 and the convergence of the corresponding H −1 gradient-flow that was proposed by Nishiura and Ohnishi [23] are also well-studied [28,12,15]. Critical points of F ε under a volume-constraint satisfy the Euler-Lagrange equation where λ ∈ R is a Lagrange-multiplier.
The energy measures µ ε as defined in (2.6) converge to a measure µ that is up to the factor 2σ integer-rectifiable and has a weak mean curvature that satisfies Finally ∂ * {u = 1} has multiplicity one H n−1 -almost everywhere in the set {v = λ} and this part of the phase boundary is a C 3,α -surface for all α < 1, except for a set of H n−1 -measure zero.
Proof. By standard elliptic theory we obtain from (7.10), (7.11) that v ε is uniformly bounded in W 2,p (Ω) for all 1 ≤ p < ∞. Therefore Theorem 3.2 applies and we can repeat the arguments of the proof of Theorem 7.1. We omit the details here.
If u ε has in addition a local energy minimizing property for F ε we can draw stronger conclusions: Then µ has multiplicity one µ-almost everywhere and is C 3,αsmooth, see the arguments in [16].
the generalized mean curvature vector of ∂ * E and define a scalar mean curvature by H := H · ∇X |∇X | on ∂ * E.