De Giorgi's inequality for the thresholding scheme with arbitrary mobilities and surface tensions
aa r X i v : . [ m a t h . A P ] J a n DE GIORGI’S INEQUALITY FOR THE THRESHOLDINGSCHEME WITH ARBITRARY MOBILITIES AND SURFACETENSIONS
TIM LAUX AND JONA LELMI
Abstract.
We provide a new convergence proof of the celebrated Merriman-Bence-Osher scheme for multiphase mean curvature flow. Our proof applies tothe new variant incorporating a general class of surface tensions and mobilities,including typical choices for modeling grain growth. The basis of the proof arethe minimizing movements interpretation of Esedo˘glu and Otto and De Giorgi’sgeneral theory of gradient flows. Under a typical energy convergence assumptionwe show that the limit satisfies a sharp energy-dissipation relation. Introduction
The thresholding scheme is a highly efficient computational scheme for multiphasemean curvature flow (MCF) which was originally introduced by Merriman, Bence,and Osher [25], [26]. The main motivation for MCF comes from metallurgy whereit models the slow relaxation of grain boundaries in polycrystals [29]. Each ”phase”in our mathematical jargon corresponds to a grain, i.e., a region of homogeneouscrystallographic orientation. The effective surface tension σ ij ( ν ) and the mobility µ ij ( ν ) of a grain boundary depend on the mismatch between the lattices of the twoadjacent grains Ω i and Ω j and on the relative orientation of the grain boundary,given by its normal vector ν . It is well known that for small mismatch angles, thedependence on the normal can be neglected [30]. The effective evolution equationsthen read V ij = − µ ij σ ij H ij along the grain boundary Σ ij , (1)where V ij and H ij denote the normal velocity and mean curvature of the grain bound-ary Σ ij = ∂ Ω i ∩ ∂ Ω j , respectively. These equations are coupled by the Herring anglecondition σ ij ν ij + σ jk ν jk + σ ki ν ki = 0 along triple junctions Σ ij ∩ Σ jk , (2)which is a balance-of-forces condition and simply states that triple junctions are inlocal equilibrium. Efficient numerical schemes allow to carry out large-scale simula-tions to give insight into relevant statistics like the average grain size or the grain boundary character distribution, as an alternative to studying corresponding meanfield limits as in [4], [16]. The main obstruction to directly discretize the dynamics(1)–(2) are ubiquitous topological changes in the network of grain boundaries likefor example the vanishing of grains. Thresholding instead naturally handles suchtopological changes. The scheme is a time discretization which alternates betweenthe following two operations: (i) convolution with a smooth kernel; (ii) thresholding.The second step is a simple pointwise operation and also the first step can be imple-mented efficiently using the Fast Fourier Transform. One of the main objectives ofour analysis is to rigorously justify this intriguingly simple scheme in the presenceof such topological changes.The basis of our analysis is the underlying gradient-flow structure of (1)–(2), whichmeans that the solution follows the steepest descent in an energy landscape. Moreprecisely, the energy is the total interfacial area weighted by the surface tensions σ ij ,and the metric tensor is the L -product on normal velocities, weighted by the inversemobilities µ ij . One can read off this structure from the inequality ddt N X i,j =1 σ ij Area(Σ ij ) = − N X i,j =1 µ ij Z Σ ij V ij dS ≤ , which is valid for sufficiently regular solutions to (1)–(2). In the seminal work[8], Esedo˘glu and Otto showed that the efficient thresholding scheme respects thisgradient-flow structure as it may be viewed as a minimizing movements scheme inthe sense of De Giorgi. More precisely, they show that each step in the scheme isequivalent to solving a variational problem of the formmin χ (cid:26) h d h (Σ , Σ n − ) + E h (Σ) (cid:27) , (3)where E h (Σ) and d h (Σ , Σ n − ) are proxies for the total interfacial energy of the con-figuration Σ and the distance of the configuration Σ to the one at the previous timestep Σ n − , respectively. Since the work of Jordan, Kinderlehrer, and Otto [14], theimportance of the formerly often neglected metric in such gradient-flow structureshas been widely appreciated. Also in the present work, the focus lies on the met-ric, which in the case of MCF is well-known to be completely degenerate [27]. Thisexplains the proxy for the metric appearing in the related well-known minimizingmovements scheme for MCF by Almgren, Taylor, and Wang, [1], and Luckhaus andSturzenhecker [23]. This remarkable connection between the numerical scheme andthe theory of gradient flows has the practical implication that it made clear how togeneralize the algorithm to arbitrary surface tensions σ ij . From the point of viewof numerical analysis, (3) means that thresholding behaves like the implicit Eulerscheme and is therefore unconditionally stability. The variational interpretation of
E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 3 the thresholding scheme has of course implications for the analysis of the algorithmas well. It allowed Otto and one of the authors to prove convergence results in themultiphase setting [17], [18], which lies beyond the reach of the more classical vis-cosity approach based on the comparison principle implemented in [9], [3], [12]. Alsoin different frameworks, this variational viewpoint turned out to be useful, such asMCF in higher codimension [21] or the Muskat problem [13]. The only downsideof the generalization [8] are the somewhat unnatural effective mobilities µ ij = σ ij .Only recently, Salvador and Esedo˘glu [31] have presented a strikingly simple way toincorporate a wide class of mobilities µ ij as well. Their algorithm is based on the factthat although the same kernel appears in the energy and the metric, each term onlyuses certain properties of the kernel, which can be tuned independently: Startingfrom two Gaussian kernels G γ and G β of different width, they find a positive linearcombination K ij = a ij G γ + b ij G β , whose effective mobility and surface tension matchthe given µ ij and σ ij , respectively. It is remarkable that this algorithm retains thesame simplicity and structure as the previous ones [26], [8]. We refer to Section 2for the precise statement of the algorithm.In the present work, we prove the first convergence result for this new generalscheme. We exploit the gradient-flow structure and show that under the naturalassumption of energy convergence, any limit of thresholding satisfies De Giorgi’sinequality, a weak notion of multiphase mean curvature flow. This assumption isinspired by the fundamental work of Luckhaus-Sturzenhecker [23] and has appearedin the context of thresholding in [17], [18]. We expected it to hold true before theonset of singularities such as the vanishing of grains. Furthermore, at least in thesimpler two-phase case, it can be verified for certain singularities [6], [5]. We wouldin fact expect this assumption to be true generically, which however seems to be adifficult problem in the multiphase case.The present work fits into the theory of general gradient flows even better than thetwo previous ones [17], [18] and crucially depends on De Giorgi’s abstract framework,cf. [2]. This research direction was initiated by Otto and the first author and appearedin the lecture notes [19]. There, De Giorgi’s inequality is derived for the simple modelcase of two phases. Here, we complete these ideas and use a careful localizationargument to generalize this result to the multiphase case. A further particular noveltyof our work is that for the first time, we prove the convergence of the new schemefor arbitrary mobilities [31].Our proof rests on the fact that thresholding, like any minimizing movementsscheme, satisfies a sharp energy-dissipation inequality of the form E h (Σ h ( T )) + 12 Z T (cid:18) h d h (Σ h ( t ) , Σ h ( t − h )) + | ∂E h | ( ˜Σ h ( t )) (cid:19) dt ≤ E h (Σ(0)) , (4) TIM LAUX AND JONA LELMI where Σ h ( t ) denotes the piecewise constant interpolation in time of our approxima-tion, ˜Σ h ( t ) denotes another, intrinsic interpolation in terms of the variational scheme,cf. Lemma 3, and | ∂E h | is the metric slope of E h , cf. (33).Our main goal is to pass to the limit in (4) and obtain the sharp energy-dissipationrelation for the limit, which in the simple two-phase case formally reads σ Area(Σ( T )) + 12 Z T Z Σ( t ) (cid:18) µ V + σ µH (cid:19) dS dt ≤ σ Area(Σ(0)) . (5)To this end, one needs sharp lower bounds for the terms on the left-hand side of (4).While the proof of the lower bound on the metric slope of the energylim inf h ↓ Z T | ∂E h | ( ˜Σ h ( t )) dt ≥ σ µ Z T Z Σ( t ) H dS dt (6)is a straight-forward generalization of the argument in [19], the main novelty of thepresent work lies in the sharp lower bound for the distance-term of the formlim inf h ↓ Z T h d h (Σ h ( t ) , Σ h ( t − h )) dt ≥ µ Z T Z Σ( t ) V dS dt. (7)This requires us to work on a mesoscopic time scale τ ∼ √ h , which is much largerthan the microscopic time-step size h and which is natural in view of the parabolicnature of our problem. It is remarkable that De Giorgi’s inequality (5) in factcharacterizes the solution of MCF under additional regularity assumptions. Indeed,if Σ( t ) evolves smoothly, this inequality can be rewritten as12 Z T Z Σ( t ) σ (cid:16) √ µσ V + √ σµH (cid:17) dS dt ≤ , (8)and therefore V = − µσH . For expository purpose, we focused here on the vanillatwo-phase case. In the multiphase case, the resulting inequality implies both thePDEs (1) and the balance-of-forces conditions (2), cf. Remark 1. An optimal energy-dissipation relation like the one here also plays a crucial role in the recent weak-stronguniqueness result for multiphase mean curvature flow by Fischer, Hensel, Simon, andone of the authors [10]. There, a new dynamic analogue of calibrations is introducedand uniqueness is established in the following two steps: (i) any strong solutionis a calibrated flow and (ii) every calibrated flow is unique in the class of weaksolutions. De Giorgi’s general strategy we are implementing here is also related tothe approaches by Sandier and Serfaty [32] and Mielke [28]. They provide sufficientconditions for gradient flows to converge in the same spirit as Γ-convergence of energyfunctionals, implies the convergence of minimizers. In the dynamic situation it isclear that one needs conditions on both energy and metric in order to verify such aconvergence. E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 5
There has been continuous interest in MCF in the mathematics literature, so weonly point out some of the most relevant recent advances. We refer the interestedreader to the introductions of [17] and [20] for further related references. The ex-istence of global solutions to multiphase MCF has only been established recentlyby Kim and Tonegawa [15] who carefully adapt Brakke’s original construction andshow in addition that phases do not vanish spontaneously. For the reader who wantsto familiarize themselves with this topic, we recommend the recent notes [34]. An-other approach to understanding the long-time behavior of MCF flow is to restartstrong solutions after singular times. This amounts to solving the Cauchy problemwith non-regular initial data, such as planar networks of curves with quadruple junc-tions. In this two-dimensional setting, this has been achieved by Ilmanen, Neves,and Schulze [11] by gluing in self-similarly expanding solutions for which it is pos-sible to show that the initial condition is attained in some measure theoretic way.Most recently, using a similar approach of gluing in self-similar solutions, but alsorelying on blow-ups from geometric microlocal analysis, Lira, Mazzeo, Pluda, Saez[22] were able to construct such strong solutions, prove stronger convergence towardsthe initial (irregular) network of curves, and classify all such strong solutions.The rest of the paper is structured as follows. In Section 2 we recall the thresh-olding scheme for arbitrary mobilitites introduced in [31], show its connection tothe abstract framework of gradient flows, and record the direct implications of thistheory. We state and discuss our main results in Section 3. Section 4 contains thelocalization argument in space, which will play a crucial role in the proofs which aregathered in Section 5. Finally, in the short Appendix, we record some basic factsabout thresholding.2.
Setup and the modified thresholding scheme
The modified algorithm.
We start by describing the algorithm proposed bySalvador and Esedo˘glu in [31]. Let the symmetric matrix σ = ( σ ij ) ij ∈ R N × N ofsurface tensions and the symmetric matrix µ = ( µ ij ) ij of mobilities be given. In thiswork we define for notational convenience σ ii = µ ii = 0. Let γ > β > A = ( − a ij ) ij ∈ R N × N and B = ( − b ij ) ij ∈ R N × N by a ij = √ π √ γγ − β ( σ ij − βµ − ij ) , (9) b ij = √ π √ βγ − β ( − σ ij + γµ − ij ) , (10)for i = j and a ii = b ii = 0. Then a ij , b ij are uniquely determined as solutions of thefollowing linear system TIM LAUX AND JONA LELMI (11) ( σ ij = a ij √ γ √ π + b ij √ β √ π ,µ − ij = a ij √ π √ γ + b ij √ π √ β . The algorithm introduced by Salvador and Esedo˘glu is as follows. Let the time stepsize h > G hγ := G ( d ) γh denotes the d -dimensional heat kernel (17)at time γh . Algorithm 1 (Modified thresholding scheme) . Given the initial partition Ω , ..., Ω N ,to obtain the partition Ω n +11 , ..., Ω n +1 N at time t = h ( n +1) from the partition Ω n , ..., Ω nN at time t = hn (1) For any i = 1 , ..., N form the convolutions φ n ,i = G hγ ∗ Ω ni , φ n ,i = G hβ ∗ Ω ni (2) For any i = 1 , ..., N form the comparison functions ψ ni = X j = i a ij φ n ,j + b ij φ n ,j . (3) Thresholding step, define Ω n +1 i := (cid:26) x : ψ ni ( x ) < min j = i ψ nj ( x ) (cid:27) . We will assume the following:The coefficients a ij , b ij satisfy the strict triangle inequality.(12) The matrices A and B are positive definite on (1 , ..., ⊥ . (13)In particular, for v ∈ (1 , ..., ⊥ we can define norms | v | A = v · A v, | v | B = v · B v. Observe that condition (12) is always satisfied if we choose γ large and β small pro-vided the surface tensions and the inverse of the mobilities satisfy the strict triangleinequality. Indeed define m σ = min i,j,k { σ ik + σ kj − σ ij } and M σ = max i,j,k { σ ik + σ kj − σ ij } , E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 7 where i, j, k range over all triples of distinct indices 1 ≤ i, j, k, ≤ N . Define m µ and M µ in a similar way. Then a computation shows that a ij and b ij satisfy the (strict)triangle inequality if(14) β < m σ M µ and γ > M σ m µ , which can always be achieved since γ > β > Lemma 1.
Let the matrix σ of the surface tensions and the matrix µ of the in-verse mobilities (for the diagonal we set inverses to be zeros) be negative definite on (1 , ..., ⊥ . Let γ > β be such that (15) γ > min i =1 ,...,N − s i max i =1 ,...,N − m i , β < max i =1 ,...,N − s i min i =1 ,...,N − m i where s i and m i are the nonzero eigenvalues of J σJ and J µ J respectively, wherethe matrix J has components J ij = δ ij − N . Then A and B are positive definite on (1 , ..., ⊥ . In particular, if we choose γ large enough and β small enough, condition (13) onthe matrices A , B is satisfied provided the matrices σ and µ are negative definiteon 1 , ..., ⊥ . By a classical result of Schoenberg [33] this is the case if and onlyif √ σ ij and 1 / √ µ ij are ℓ embeddable. In particular, this holds for the choice ofRead-Schockley surface tensions and equal mobilities.For 1 ≤ i = j ≤ N define the kernels(16) K ij ( z ) = a ij G γ ( z ) + b ij G β ( z )where, for a given t >
0, we define G ( d ) t as the heat kernel in R d , i.e.,(17) G ( d ) t ( z ) = e − | z | t √ πt d . If the dimension d is clear from the context, we suppress the superscript ( d ) in (17).We recall here some basic properties of the heat kernel. G t ( z ) > , (18) G t ( z ) = G t ( Rz ) ∀ R ∈ O ( d ) (symmetry) , (19) TIM LAUX AND JONA LELMI G t ( z ) = 1 √ t d G (cid:18) z √ t (cid:19) (scaling) , (20) G t ∗ G s = G t + s (semigroup property) , (21) G ( d ) t ( z ) = d Y i =1 G (1) t ( z i ) (factorization property) . (22)We observe that the kernels K ij are positive, with positive Fourier transform ˆ K ij provided γ > max i,j σ i,j µ i,j and β < min i,j σ i,j µ i,j . In particular assuming(1) σ ij and µ ij satisfy the strict triangle inequality,(2) σ and µ are negative definite on (1 , ..., ⊥ ,we can always achieve the conditions posed on A , B and the positivity of the kernels K ij by choosing γ large and β small.Given any h > K hij ( z ) = 1 √ h d K ij ( z √ h ) , then the first and the second step in Algorithm 1 may be compactly rewritten asfollows ψ ni = X j = i K hij ∗ Ω nj . For later use, we also introduce the kernel(24) K ( z ) = 12 G γ ( z ) + 12 G β ( z ) . Connection to De Giorgi’s minimizing movements.
The first observationis that Algorithm 1 has a minimizing movements interpretation. To explain this, letus introduce the class A := ( χ : [0 , d → { , } N (cid:12)(cid:12)(cid:12)(cid:12) N X k =1 χ k = 1 ) and its relaxation M := ( u : [0 , d → [0 , N (cid:12)(cid:12)(cid:12)(cid:12) N X k =1 u k = 1 ) . E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 9 If χ ∈ A ∩ BV ([0 , d ) N , then each of the sets Ω i := { χ i = 1 } is a set of finiteperimeter. We denote by ∂ ∗ Ω i the reduced boundary of the set Ω i , and for any pair1 ≤ i = j ≤ N we denote by Σ ij := ∂ ∗ Ω i ∩ ∂ ∗ Ω j the interface between the sets. For u ∈ M we define(25) E ( u ) := (P i,j σ ij H d − (Σ ij ) if u ∈ A ∩ BV ([0 , d ) N + ∞ otherwise . For h > E h for u ∈ M (26) E h ( u ) = X i,j √ h Z [0 , d u i K hij ∗ u j dx. For u, v ∈ M and h > d h ( u, v ) := − hE h ( u − v ) = − √ h X i,j Z ( u i − v i ) K hij ∗ ( u j − v j ) dx (27) = 2 √ h Z | G h/ γ ∗ ( u − v ) | A + | G h/ β ∗ ( u − v ) | B dx, where we used the semigroup property (21) and the symmetry (19) to derive the lastequality. Lemma 2.
The pair ( M , d h ) is a compact metric space. The function E h is con-tinuous with respect to d h . For every ≤ i ≤ N and n ∈ N define χ ni = Ω ni ,where Ω n , ..., Ω nN are obtained from Ω n − , ..., Ω n − N by the thresholding scheme. Then χ n minimizes (28) 12 h d h ( u, χ n − ) + E h ( u ) among all u ∈ M . Proof.
For u, v ∈ M definition (27) and the fact that A and B are positive definiteimply that d h is a distance on M . The fact that ( M , d h ) is compact and E h iscontinuous is just a consequence of the fact that d h metrizes the weak convergencein L on M , the interested reader may find the details of the reasoning in [19]. Weare thus left with showing that χ n satisfies (28). For u, v ∈ L ([0 , d ) define( u, v ) = 1 √ h X i,j Z u i K hij ∗ v j dx, then by the symmetry (19) of the Gaussian kernel and by the symmetry of bothmatrices A , B it is not hard to show that ( · , · ) is symmetric. In particular we canwrite for any u ∈ M h d h ( u, χ n − ) + E h ( u ) = − E h ( u − χ n − ) + E h ( u )= − ( u − χ n − , u − χ n − ) + ( u, u )= 2( χ n − , u ) − ( χ n − , χ n − ) . Thus (28) is equivalent to the fact that χ n minimizes ( χ n − , u ) among all u ∈ M .Since by (12) ( χ n − , u ) = Z X i u i ψ ni dx, we see that χ n minimizes the integrand pointwise, and thus it is a minimizer for thefunctional. (cid:3) The previous lemma allows us to apply the general theory of gradient flows in [2]to this particular problem. We record the key statement for our purposes in thefollowing lemma.
Lemma 3.
Let ( M , d ) be a compact metric space and E : M → R be continuous.Given χ ∈ M and h > consider a sequence { χ n } n ∈ N satisfying (29) χ n minimizes h d ( u, χ n − ) + E ( u ) among all u ∈ M . Then we have for all t ∈ N hE ( χ ( t )) + 12 Z t (cid:18) h d ( χ ( s + h ) , χ ( s )) + | ∂E | ( u ( s )) (cid:19) ds ≤ E ( χ ) . (30) Here χ ( t ) is the piecewise constant interpolation, u ( t ) is another interpolationsatisfying Z ∞ h d ( u ( t ) , χ ( t )) dt ≤ E ( χ ) , (31) E ( u ( t )) ≤ E ( χ ( t )) for all t ≥ , (32) and | ∂E | ( u ) is the metric slope defined by (33) | ∂E | ( u ) := lim d ( u,v ) → ( E ( u ) − E ( v )) + d ( u, v ) ∈ [0 , ∞ ] . E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 11 Statement of results
Our main result is the convergence of the modified thresholding scheme to a weaknotion of multiphase mean curvature flow. More precisely, given an initial partition { Ω , ..., Ω N } of [0 , d encoded by χ : [0 , d → { , } N such that P i χ i = 1, define χ h : [0 , d × R → { , } N by setting χ h ( t, x ) = χ ( x ) for t < h,χ h ( t, x ) = χ n ( x ) for t ∈ [ nh, ( n + 1) h ) for n ∈ N . (34)If χ is a function of bounded variation, we denote by Σ ij := ∂ ∗ Ω i ∩ ∂ ∗ Ω j . Ourmain result is contained in the following theorem. Theorem 1.
Given χ ∈ A and such that ∇ χ is a bounded measure and a sequence h ↓ ; let χ h be defined by (34). Assume that there exists χ : [0 , d × (0 , T ) → [0 , N such that (35) χ h ⇀ χ in L ([0 , d × (0 , T )) . Then χ ∈ { , } N almost everywhere, P i χ i = 1 and χ ∈ L ((0 , T ) , BV ([0 , d )) N .If we assume that (36) lim sup h ↓ Z T E h ( χ h ( t )) dt ≤ X ij σ ij Z T H d − (Σ ij ( t )) dt, then χ is a De Giorgi solution in the sense of Definition 1 below. Definition 1.
Given χ ∈ A and such that ∇ χ is a bounded measure, a map χ : [0 , d × (0 , T ) → { , } N such that P i χ i = 1 and χ ∈ L ((0 , T ) , BV ([0 , d )) N is called a De Giorgi solution to the multiphase mean curvature flow with surfacetensions σ ij and mobilities µ ij provided the following three facts hold:(1) There exist H ij ∈ L ( H d − | Σ ij ( dx ) dt ) which are mean curvatures in the weaksense, i.e., such that for any test vector field ξ ∈ C ∞ c ([0 , d × (0 , T )) d X i,j σ ij Z [0 , d × (0 ,T ) ( ∇ · ξ − ν ij · ∇ ξν ij ) H d − | Σ ij ( t ) ( dx ) dt (37) = − X i,j σ ij Z [0 , d × (0 ,T ) H ij ν ij · ξ H d − | Σ ij ( t ) ( dx ) dt. (2) There exist normal velocities V ij ∈ L ( H d − | Σ ij ( t ) ( dx ) dt ) with Z [0 , d η ( t = 0) χ i dx + Z [0 , d × (0 ,T ) ∂ t η χ i dxdt + X k = i Z [0 , d × (0 ,T ) ηV ik H d − | Σ ik ( t ) ( dx ) dt = 0for all η ∈ C ∞ c ([0 , d × [0 , T )).(3) De Giorgi’s inequality is satisfied, i.e.lim sup τ ↓ τ X i,j σ ij Z ( T − τ,T ) H d − (Σ ij ( t )) dt + 12 X i,j Z [0 , d × (0 ,T ) (cid:18) V ij µ ij + σ ij µ ij H ij (cid:19) H d − | Σ ij ( t ) ( dx ) dt ≤ X i,j σ ij H d − (Σ ij ) . (38) Remark . Observe that inequality (38) together with the definition of the weak meancurvatures gives a notion of weak solution for the multiphase mean curvature flowincorporating both the dynamics V ij = − σ ij µ ij H ij and the Herring angle conditionat triple junctions. Indeed if χ : [0 , d × (0 , T ) → { , } N with P i χ i ( t ) = 1 is suchthat the sets Ω i ( t ) = { χ i ( · , t ) = 1 } meet along smooth interfaces Σ ij := ∂ Ω i ∩ ∂ Ω j which evolve smoothly and satisfy (37), (38) then(1) The Herring angle condition at triple junctions is satisfied. Indeed by thedivergence theorem on surfaces (see Theorem 11.8 and Remark 11.42 in [24])for any ξ ∈ C ∞ c ( × [0 , d ) d Z Σ ij ( t ) ( ∇ · ξ − ν ij · ∇ ξν ij ) H d − | Σ ij ( t ) ( dx ) = − Z Σ ij ( t ) H ij ν ij H d − | Σ ij ( t ) ( dx )+ Z ∂ Σ ij ( t ) ξ · ν ij H d − ( dx ) . Thus (37) and H ij ∈ L ( H d − | Σ ij ( t ) ( dx ) dt ) imply that σ i i Z ∂ Σ i i ( t ) ξ · ν i i H d − ( dx )+ σ i i Z ∂ Σ i i ( t ) ξ · ν i i H d − ( dx )+ σ i i Z ∂ Σ i i ( t ) ξ · ν i i H d − ( dx ) = 0 , which forces σ i i ν i i + σ i i ν i i + σ i i ν i i = 0 at triple junctions. E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 13 (2) We have V ij = − σ ij µ ij H ij on Σ ij ( t ). Indeed in the smooth case inequality(38) reduces to X i,j σ ij Z (0 ,T ) ddt H d − (Σ ij ( t )) dt + 12 X i,j Z [0 , d × (0 ,T ) (cid:18) V ij µ ij + σ ij µ ij H ij (cid:19) H d − | Σ ij ( t ) ( dx ) dt ≤ . If one recalls that ddt H d − (Σ ij ( t )) = R [0 , d V ij H ij H d − | Σ ij ( t ) ( dx ), after completingthe square we arrive at X i,j Z [0 , d × (0 ,T ) (cid:18) V ij √ µ ij + σ ij µ ij H ij (cid:19) H d − | Σ ij ( t ) ( dx ) dt ≤ , which implies V ij = − σ ij µ ij H ij .The following lemma establishes, next to a compactness statement, that our con-vergence can be localized in the space and time variables x and t , but also in thevariable z appearing in the convolution. Lemma 4. (1)
Let { χ h } h ↓ be a sequence of { , } N -valued functions on (0 , T ) × [0 , d that satisfies (39) lim sup h ↓ (cid:18) esssup t ∈ (0 ,T ) E h ( χ h ( t )) + Z T h d h ( χ h ( t ) , χ h ( t − h )) dt (cid:19) < ∞ and that is piecewise constant in time in the sense of (34). Such a sequenceis compact in L ([0 , d × (0 , T )) N and any weak limit χ is such that χ ∈ L ((0 , T ) , BV ([0 , d )) N with (40) X i,j σ ij Z T H d − (Σ ij ( t )) dt ≤ lim inf h ↓ Z T E h ( χ h ( t )) dt. (2) Assume that u h is a sequence of [0 , N -valued functions with P i u hi = 1 suchthat (36) holds (with χ h replaced by u h ) and such that u h → χ in L ([0 , d × (0 , T )) N holds. Assume also that (41) lim sup h ↓ esssup t ∈ (0 ,T ) E h ( u h ( t )) < ∞ . Then as measures on R d × [0 , d × (0 , T ) we have the following weak conver-gences for any i = j K ij ( z ) √ h u hi ( x, t ) u hj ( x − √ hz, t ) dxdtdz⇀ K ij ( z )( ν ij ( x, t ) · z ) + H d − | Σ ij ( t ) ( dx ) dtdz. (42) K ij ( z ) √ h u hi ( x − √ hz, t ) u hj ( x, t ) dxdtdz⇀ K ij ( z )( ν ij ( x, t ) · z ) − H d − | Σ ij ( t ) ( dx ) dtdz. (43) Here the convergence may be tested also with continuous functions which havepolynomial growth in z ∈ R d . The next proposition is the main ingredient in the proof of Theorem 1. It estab-lishes the sharp lower bound on the distance-term.
Proposition 1.
Suppose that (35) and the conclusion of Lemma 4 (2) hold. Assumealso that the left hand side of (45) is finite. Then for every ≤ k ≤ N there exists V k ∈ L ( |∇ χ k | dt ) such that (44) ∂ t χ k = V k |∇ χ k | dt in the sense of distributions. Given i = j , it holds that V i ( x, t ) = − V j ( x, t ) on Σ ij ( t ) and if we define V ij ( x, t ) := V i ( x, t ) ν ij ( x, t ) | Σ ij ( t ) then we have (45) lim inf h ↓ Z T h d h ( χ h ( t ) , χ h ( t − h )) dt ≥ X i,j µ ij Z T Z Σ ij ( t ) | V ij ( x, t ) | H d − ( dx ) dt. The final ingredient is the analogous sharp lower bound for the metric slope.
Proposition 2.
Suppose that the conclusion of Lemma 4 (2) holds and that (35)holds with χ h replaced by u h . Then for any i = j there exists a mean curvature H ij ∈ L ( H d − ij ( t ) ( dx ) dt ) in the sense of (37). Moreover the following inequality istrue: lim inf h ↓ Z T | ∂E h | ( u h ( t )) dt ≥ X i,j µ ij σ ij Z T Z Σ ij ( t ) | H ij ( x, t ) | H d − ( dx ) dt. Construction of suitable partitions of unity
In the sequel we will frequently want to localize on one of the interfaces. To do so,we need to construct a suitable family of balls on wich the behavior of the flow is splitinto two majority phases and several minority phases. Hereafter we will ignore the
E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 15 time variable and consider a map χ : [0 , d → { , } N such that χ ∈ BV ([0 , d , R N ), P k χ k = 1. Given 1 ≤ i < j ≤ N we denote by ∂ ∗ Ω i the reduced boundary of theset { χ i = 1 } and by Σ ij = ∂ ∗ Ω i ∩ ∂ ∗ Ω j the interface between phase i and phase j .Given a real number r > n ∈ N we define(46) F rn := n B ( x, nr √ d ) : x ∈ r Z d o where the balls appearing in the definition are intended to be open. Observe thatfor any n ≥ r > F rn is a covering of R d withthe property that any point x ∈ R d lies in at most c ( n, d ) distinct balls belongingto F rn , where 0 < c ( n, d ) ≤ (2 n ) d is a constant that depends on n, d but not on r .Given numbers 1 ≤ l = p ≤ N we define(47) E r := (cid:26) B ∈ F r : B ∩ Σ lp = ∅ , H d − (Σ ij ∩ B ) ω d − (4 r ) d − ≤ d , { i, j } 6 = { l, p } (cid:27) . Here 2 B denotes the ball with center given by the center of B and twice its radius.Given l, p as above, denote by { B rm } m ∈ N an enumeration of E r and by { ρ m } m ∈ N asmooth partition of unity subordinate to { B rm } m ∈ N . Lemma 5.
Fix ≤ l = p ≤ N . With the above construction the following twoproperties hold. (1) For any ≤ i = j ≤ N , { i, j } 6 = { l, p } and any η ∈ L ( H d − | Σ ij )(48) lim r ↓ X m ∈ N Z B rm η H d − | Σ ij ( dx ) = 0 . (2) For any η ∈ L ( H d − | Σ lp )(49) lim r ↓ X m ∈ N Z ρ m η H d − | Σ lp ( dx ) = Z η H d − | Σ lp ( dx ) . Proofs
Proof of Theorem 1.
By Lemma 2, we can apply Lemma 3 on the metric space( M , d h ) so that we get inequality (30) with ( E, d, χ, u ) = ( E h , d h , χ h , u h ). Our firstobservation is that(50) lim h ↓ E h ( χ ) = X i,j σ ij H d − (Σ ij ) , which follows from the consistency, cf. Lemma 7 in the Appendix. Inequality (30)then yields that the sequence χ h satisfies (39), so that Lemma 4 (1) applies to getthat χ ∈ L ((0 , T ) , BV ([0 , d )) N , χ ∈ { , } N a.e., P i χ i = 1 and, after extractinga subsequence, χ h → χ in L ([0 , d × (0 , T )) N . We claim that this implies u h → χ in L ([0 , d × (0 , T )) N . To see this, observe that (31) implies hE h ( χ ) ≥ − Z T E h ( u h ( t ) − χ h ( t )) dt ≥ C √ h N X i =1 (cid:18)Z | G h/ γ ∗ ( u hi − χ hi ) | dxdt + Z | G h/ β ∗ ( u hi − χ hi ) | dxdt (cid:19) (51)where C is a constant which depends on N, A , B but not on h and comes from thefact that all norms on (1 , ..., ⊥ are comparable. Inequality (51) clearly implies that K h ∗ u h − K h ∗ χ h converges to zero in L . Observe that inequality (32) in particularyields (41) with χ h replaced by u h . Recalling (152) in the Appendix, we learn that u h − χ h converges to zero in L . This implies that we can apply Lemma 4 (2) bothto the sequence u h and the sequence χ h . In particular, we may apply Proposition 1for χ h and Proposition 2 for u h . Now the proof follows the same strategy as the onein the two-phase case in [19]. For the sake of completeness, we sketch the argumenthere. First of all, Lemma 3 gives inequality (30) for ( E h , d h , χ h , u h ), namely for n ∈ N ρ ( nh ) ≤ E h ( χ ) , (52)where we set ρ ( t ) = E h ( χ h ( t )) + R t (cid:0) h d h ( χ h ( s + h ) , χ h ( s )) + | ∂E h ( u h ( s )) | s (cid:1) ds .Multiplying (52) by η ( nh ) − η (( n + 1) h ) for some non-increasing function η ∈ C c ([0 , T )) we get − R dηdt ρdt ≤ ( η (0) + h sup (cid:12)(cid:12) dηdt (cid:12)(cid:12) ) E h ( χ ). As test function η , wenow choose η ( t ) = max { min { T − tτ , } , } and obtain1 τ Z TT − τ E h ( χ h ( t )) dt (53) + 12 Z T − τ (cid:18) h d h ( χ h ( t ) , χ h ( t − h )) + | ∂E h ( u h ( t )) | (cid:19) dt ≤ (1 + hτ ) E h ( χ ) . Now it remains to pass to the limit as h ↓
0: to get (38) from inequality (53) oneuses the lower semicontinuity (40) for the first left hand side term, the sharp bound(45) for the second left hand side term, the bound (46) for the last left hand sideterm and finally one uses the consistency Lemma 7 in the Appendix to treat theright hand side term. To get (38) it remains to pass to the limit in τ ↓ E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 17 (cid:3)
Proof of Lemma 4.
Argument for (1). For the compactness, the arguments in [19]adapt to this setting with minor changes. The first observation is that, by inequality(152) in the Appendix, one needs to prove compactness in L ([0 , d × (0 , T )) N of { K h ∗ χ h } h ↓ . For this, one just needs a modulus of continuity in time. I.e. it issufficient to prove that there exists a constant C > h such that I h ( s ) ≤ C √ s , where I h ( s ) = Z ( s,T ) × [0 , d | χ h ( x, t ) − χ h ( x, t − s ) | dxdt. This is can be done applying word by word the argument in [19] once we show thefollowing: for any pair χ, χ ′ ∈ A of admissible functions, we have(54) Z | χ − χ ′ | dx ≤ C √ h d h ( χ, χ ′ ) + C √ h ( E h ( χ ) + E h ( χ ′ )) . Here the constant C depends on N, A , B but not on h .To prove (54) we proceed as follows: let S ∈ R N × N be a symmetric matrix whichis positive definite on (1 , ..., ⊥ . Since any two norms on a finite dimensional spaceare comparable, there exists a constant C > S and N such that(55) | χ − χ ′ | ≤ | χ − χ ′ | ≤ C | χ − χ ′ | S where | · | S denotes the norm induced by S . For a function u ∈ M write ( ˜ K h ∗ ) u h forthe function (cid:16) ( ˜ K h ∗ ) u h (cid:17) i = X j = i K hij ∗ u jh . Then we calculate | χ − χ ′ | S = − ( χ − χ ′ ) · ( ˜ K h ∗ )( χ − χ ′ ) + ( χ − χ ′ )( S + ( ˜ K h ∗ ))( χ − χ ′ ) . (56)Select S = ( s ij ) where s ij = − R K ij ( z ) dz . Then, by our assumption (13) S is positivedefinite on (1 , ..., ⊥ and after integration on [0 , d identity (56) becomes Z | χ − χ ′ | S dx = 12 √ h d h ( χ, χ ′ ) + Z ( χ − χ ′ )( S + ( ˜ K h ∗ ))( χ − χ ′ ) dx. We now proceed to estimate the integral on the right hand side. By the choice of S and Jensen’s inequality we have Z ( χ − χ ′ )( S + ( ˜ K h ∗ ))( χ − χ ′ ) dx ≤ C Z | S + ( ˜ K h ∗ ))( χ − χ ′ ) | dx (57) ≤ C X i,j Z K hij ( z ) | ( χ j − χ ′ j )( x − z ) − ( χ j − χ ′ j )( x ) | dxdz. Using the triangle inequality and (150) in the Appendix we can estimate the righthand side to obtain the following inequality Z ( χ − χ ′ )( S + ( ˜ K h ∗ ))( χ − χ ′ ) dx (58) ≤ C X i,j X k = j Z K hij ( z ) χ j ( x − z ) χ k ( x ) dxdz + X k = j Z K hij ( z ) χ j ( x ) χ k ( x − z ) dxdz + X k = j Z K hij ( z ) χ ′ j ( x − z ) χ ′ k ( x ) dxdz + X k = j Z K hij ( z ) χ ′ j ( x ) χ ′ k ( x − z ) dxdz ! . Observing that there is a constant
C > K ij ≤ CK jk we conclude that Z ( χ − χ ′ )( S + ( ˜ K h ∗ ))( χ − χ ′ ) dx ≤ C √ h ( E h ( χ ) + E h ( χ ′ )) . This proves (54) and closes the argument for the compactness.We also have to prove (40), but this follows from (42) with u h replaced by χ h oncewe have shown that the limit χ is such that |∇ χ | is a bounded measure, equiintegrablein time. This can be done with an argument similar to the one used in [19] for thetwo-phase case. Observe that this only requires the weaker assumption (41).Argument for (2). As mentioned in the previous paragraph, we already know thatthe limit χ is such that |∇ χ | is a bounded measure, equiintegrable in time. We willprove (42). Then (43) easily follows by recalling that ν ij = − ν ji . A standard argu-ment (to be found in [19]) which relies on the exponential decay of the kernel yieldsthe fact that we can test convergences (42) with functions with at most polynomialgrowth in z provided we already have the result for bounded and continuous testfunctions, thus we focus on this case. E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 19
Let ξ ∈ C b ( R d × [0 , d × (0 , T )) be a bounded and continuous function. To show(42) we aim at showing thatlim h ↓ Z ξ ( z, x, t ) K ij ( z ) √ h u hi ( x, t ) u hj ( x − √ hz, t ) dxdtdz = Z ξ ( z, x, t ) K ij ( z )( ν ij ( x, t ) · z ) + H d − ij ( t ) ( dx ) dtdz. (59)Upon splitting ξ into the positive and the negative part, by linearity we mayassume that 0 ≤ ξ ≤
1. We can split (59) into the local lower boundlim inf h ↓ Z ξ ( z, x, t ) K ij ( z ) √ h u hi ( x, t ) u hj ( x − √ hz, t ) dzdxdt ≥ Z ξ ( z, x, t ) K ij ( z )( ν ij ( x, t ) · z ) + H d − ij ( t ) ( dx ) dtdz. (60)and the global upper boundlim inf h ↓ Z K ij ( z ) √ h u hi ( x, t ) u hj ( x − √ hz, t ) dzdxdt ≤ Z K ij ( z )( ν ij ( x, t ) · z ) + H d − ij ( t ) ( dx ) dtdz. (61)Indeed we can recover the limsup inequality in (59) by splitting ξ = 1 − (1 − ξ ) andapplying the local lower bound (60) to 1 − ξ .We first concentrate on the local lower bounds in the case where u h = χ , namelywe will showlim inf h ↓ Z ξ ( z, x, t ) K ij ( z ) √ h χ i ( x, t ) χ j ( x − √ hz, t ) dzdxdt ≥ Z ξ ( z, x, t ) K ij ( z )( ν ij ( x, t ) · z ) + H d − ij ( t ) ( dx ) dtdz. (62)By Fatou’s lemma the claim is reduced to showing that for a.e. point t in time andevery z ∈ R d lim inf h ↓ Z ξ ( z, x, t ) K ij ( z ) √ h χ i ( x, t ) χ j ( x − √ hz, t ) dx (63) ≥ Z ξ ( z, x, t ) K ij ( z )( ν ij ( x, t ) · z ) + H d − ij ( t ) ( dx ) . Fix a point t such that χ ( · , t ) ∈ BV ([0 , d , { , } N ) and any z ∈ R d . In the sequel,we will drop those variables, so χ ( x ) = χ ( x, t ), ξ ( x ) = ξ ( z, x, t ). By approximationwe may assume that ξ ∈ C ∞ ([0 , d ). Let ρ m be a partition of unity obtained byapplying the construction of Section 4 to the function χ ( x ) on the interface Σ ij . Thenby Lemma 5 we have Z ξ ( x )( ν ij ( x ) · z ) + H d − | Σ ij ( dx )= lim r ↓ X m ∈ N Z ρ mij ( x ) ξ ( x )( ν ij ( x ) · z ) + H d − | Σ ij ( dx ) ! = lim r ↓ X m ∈ N (cid:18)Z ρ mij ( x ) ξ ( x )( ν i ( x ) · z ) + H d − | ∂ ∗ Ω i ( dx ) − X k = i,j Z ρ mij ( x ) ξ ( x )( ν ij ( x ) · z ) + H d − | Σ ik ( dx ) ! = lim r ↓ X m ∈ N Z ρ mij ( x ) ξ ( x )( ν i ( x ) · z ) + H d − | ∂ ∗ Ω i ( dx ) . (64)We now focus on estimating the argument of the last limit. Observe that ( ν ij ( x ) · z ) + H d − | ∂ ∗ Ω i ( dx ) = ( ∂ z χ i ) + , thus by definition of positive part of a measure, given ǫ > m ∈ N , a function ˜ ξ m ∈ C c ( B m ) such that 0 ≤ ˜ ξ m ≤ Z ρ mij ξ ˜ ξ m ∂ z χ i + 2 − m ǫ ≥ Z ρ mij ξ ( ν i · z ) + H d − | ∂ ∗ Ω i ( dx ) . Let η m := ρ mij ξ ˜ ξ m ∈ C c ( B m ), then Z η m ∂ z χ i = − Z ∂ z η m χ i dx = lim h ↓ Z η m ( x + √ hz ) − η ( x ) √ h χ i ( x ) dx = lim h ↓ Z η m ( x ) χ i ( x ) − χ i ( x − √ hz ) √ h dx ≤ lim inf h ↓ X k = i Z η m ( x ) χ i ( x ) χ k ( x − √ hz ) √ h dx E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 21 ≤ lim inf h ↓ Z η m ( x ) χ i ( x ) χ j ( x − √ hz ) √ h dx + lim sup h ↓ X k = i,j Z η m ( x ) χ i ( x ) χ k ( x − √ hz ) √ h dx ≤ lim inf h ↓ Z η m ( x ) χ i ( x ) χ j ( x − √ hz ) √ h dx + X k = i,j lim sup h ↓ Z η m ( x ) χ i ( x ) χ k ( x − √ hz ) √ h dx. Observe that for each m ∈ N , using also the consistency Lemma 7lim sup h ↓ Z η m ( x ) χ i ( x ) χ k ( x − √ hz ) √ h dx ≤ lim sup h ↓ Z η m ( x ) χ i ( x ) χ k ( x − √ hz ) + χ i ( x − √ hz ) χ j ( x ) √ h dx = Z η m ( x ) | ν ik ( x ) · z |H d − | Σ ik ( dx ) ≤ | z |H d − ( B rmij ∩ Σ ik )Thus we obtain Z η m ∂ z χ i ≤ lim inf h ↓ Z η m ( x ) χ i ( x ) χ j ( x − √ hz ) √ h dx + X k = i,j | z |H d − ( B rmij ∩ Σ ik )Inserting back into (64), recalling also Lemma 5 and the inequality (65), usingFatou’s lemma, the fact that ρ mij is a partition of unity and that 0 ≤ ˜ ξ m ≤ Z ξ ( x )( ν ij ( x ) · z ) + H d − | Σ ij ( dx ) ≤ lim inf h ↓ Z ξ ( x ) χ i ( x ) χ j ( x − √ hz ) √ h dx + ǫ and (62) follows letting ǫ go to zero. To derive inequality (60) we just apply Lemma9 in the Appendix.To get the upper bound (61) we argue as follows. First of all recall Assumption(36) which says (66) Z T E h ( u h ( t )) dt → Z T E ( χ ( t )) dt. Now, if we define e ijh ( u h ) = 1 √ h Z T Z u hi ( t ) K hij ∗ u hj ( t ) dxdt we have that by (60) lim inf h ↓ e ijh ( u h ) ≥ e ij ( χ ), where e ij ( χ ) is defined in the obviousway. Assume that there exists a pair i, j such that lim sup h ↓ e ijh ( u h ) > e ij ( χ ), then Z T E ( χ ( t )) dt = lim h ↓ Z T E h ( u h ( t )) dt = lim sup h ↓ Z T E h ( u h ( t )) dt ≥ X ( l,p ) =( i,j ) lim inf h ↓ e lph ( u h ) + lim sup h ↓ e ijh ( u h ) > Z T E ( χ ( t )) dt which is a contradiction. Thus we have proved (61). (cid:3) Proof of Proposition 1.
Since we assume that the left hand side of (45) is finite, inview of (27), upon passing to a subsequence we may assume that, in the sense ofdistributions, the limit(67) lim h ↓ h √ h (cid:18)(cid:12)(cid:12) G h/ γ ∗ ( χ − χ ( · − h )) (cid:12)(cid:12) A + (cid:12)(cid:12)(cid:12) G h/ β ∗ ( χ − χ ( · − h )) (cid:12)(cid:12)(cid:12) B (cid:19) = ω exists as a finite positive measure on [0 , d × (0 , T ). Here we indicated with χ hl ( · − h )the time shift of function χ hl . We denote by τ a small fraction of the characteristicspatial scale, namely τ = α √ h for some α >
0, which we think as a small number.Given 1 ≤ l ≤ N we define(68) δχ hl := χ hl − χ hl ( · − τ ) . We divide the proof into two parts: first we show that the normal velocities exist,and afterwards we prove the sharp bound. But first, let us state two distributionalinequalities that will be used later. Namely
E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 23 • In a distributional sense it holds that(69) lim sup h ↓ − √ h X i = j δχ i K hij ∗ δχ j ≤ α ω. • There exists a constant
C > ≤ i ≤ N and any θ ∈ { γ, β } in a distributional sense it holds that(70) lim sup h ↓ √ h ( χ i − χ i ( · − τ )) G hθ ∗ ( χ i − χ i ( · − τ )) ≤ Cα ω. We observe that it suffices to prove (69), then (70) follows immediately. Indeedrecall that A and B are positive definite on (1 , ..., ⊥ . In particular there exists aconstant C > v ∈ (1 , ..., ⊥ one has | v | A + | v | B ≥ C | v | ≥ Cv i for any i ∈ { , ..., N } . Applying this to the vector v = G h/ θ ∗ δχ i one gets(71) | G h/ θ ∗ δχ i | ≤ C | G h/ θ ∗ δχ | A + | G h/ θ ∗ δχ | B . The claim then follows from the definition of ω , (69), the symmetry (19) and thesemigroup property (21). Indeed it is sufficient to check that, in the sense of distri-butions(72) lim h ↓ √ h X i = j δχ i K hij ∗ δχ j + 1 √ h (cid:16) | G h/ γ ∗ δχ | A + | G h/ β ∗ δχ | B (cid:17) = 0 . To this aim, pick a test function η ∈ C ∞ c ([0 , d × (0 , T )). Spelling out the definitionof the norms | · | A and | · | B , the claim is proved once we show thatlim h ↓ √ h X i,j a ij Z ξ ( δχ i G hγ ∗ δχ j − G h/ γ ∗ δχ i G h/ γ ∗ δχ j ) dxdt = 0 , (73)and the same claim with a ij , γ replaced by b ij , β respectively.We concentrate on (73). Clearly, we are done once we show that for any i = j lim h ↓ √ h Z ξ ( δχ i G hγ ∗ δχ j − G h/ γ ∗ δχ i G h/ γ ∗ δχ j ) dxdt = 0 . (74)To show this, using the semigroup property (21) we rewrite the argument of the limitas (75) − √ h Z [ ξ, G h/ γ ∗ ]( δχ i ) G h/ γ ∗ δχ j dxdt, and we observe that by the boundedness of the measures √ h | G h/ γ ∗ δχ | A it sufficesto show(76) lim h ↓ √ h Z | [ ξ, G h/ γ ∗ ]( δχ i ) | dxdt = 0 . To prove this, spelling out the integrand, using the Cauchy-Schwarz inequality andrecalling the scaling (20) we observe that Z | [ ξ, G h/ γ ∗ ]( δχ i ) | dxdt ≤ Z (cid:18)Z | ξ ( x, t ) − ξ ( x − z, t ) | G h/ γ ( z ) dz (cid:19) G h/ γ ∗ | δχ i ( x, t ) | dxdt ≤ h |∇ ξ | Z G γ ( z ) | z | dz Z T Z | δχ i ( x, t ) | dxdt. (77)Observe that by the compactness of χ h in L ([0 , d × (0 , T )), (77) is of order h , thus(76) indeed holds true.The proof of (69) is essentially already contained in the paper [19]. For the conve-nience of the reader we sketch the main ideas here. One reduces the claim to provingthe following facts.(78) lim h ↓ √ h X ij δχ i K hij ∗ δχ j − √ h (cid:18)(cid:12)(cid:12) G h/ γ ∗ δχ (cid:12)(cid:12) A + (cid:12)(cid:12)(cid:12) G h/ β ∗ δχ (cid:12)(cid:12)(cid:12) B (cid:19) = 0 . (79) lim sup h ↓ √ h (cid:12)(cid:12) G h/ γ ∗ δχ (cid:12)(cid:12) A − α h √ h (cid:12)(cid:12) G h/ γ ∗ ( χ − χ ( · − h )) (cid:12)(cid:12) A ≤ . (80) lim sup h ↓ √ h (cid:12)(cid:12)(cid:12) G h/ β ∗ δχ (cid:12)(cid:12)(cid:12) B − α h √ h (cid:12)(cid:12)(cid:12) G h/ β ∗ ( χ − χ ( · − h )) (cid:12)(cid:12)(cid:12) B ≤ . Claim (78) was proved in the previous paragraph, while (79) and (80) are conse-quences of Jensen’s inequality in the time variable for the convex functions | · | A and | · | B respectively. More precisely, assume without loss of generality that τ = N h forsome N ∈ N , then by a telescoping argument and Jensen’s inequality for | · | A we get E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 25 √ h | G h/ γ ∗ δχ | A ≤ N N − X n =0 √ h | G h/ γ ∗ ( χ h ( · − nh ) − χ h ( · − ( n + 1) h )) | A . Recalling that N = α/ √ h we can rewrite the right hand side as(81) α N N − X n =0 h √ h | G h/ γ ∗ ( χ h ( · − nh ) − χ h ( · − ( n + 1) h )) | A . This is an average of time shifts of α h √ h | G h/ γ ∗ ( χ h − χ h ( · − h )) | A . Since N h = o (1)all these time shifts are small, thus the average has the same distributional limit as α h √ h | G h/ γ ∗ ( χ h − χ h ( · − h )) | A . This proves (79). The argument for (80) is similar. Existence of the normal velocities.
We now prove the existence of the normalvelocities. Fix 1 ≤ i ≤ N and observe that for w ∈ { γ, β } we have | χ i − χ i ( − τ ) | ≤ ( χ i − χ i ( − τ )) G hw ∗ ( χ i − χ i ( − τ )) + | χ i − G hw ∗ χ i | + | χ i ( − τ ) − G hw ∗ χ i ( − τ ) | , (82)which follows simply by observing that | χ i − χ i ( · − τ ) | = | χ i − χ i ( · − τ ) | = ( χ i − χ i ( · − τ ) G hw ∗ ( χ i − χ i ( · − τ )) + ( χ i − χ i ( · − τ ))( χ − G hw ∗ χ ) + ( χ i ( · − τ ) − χ i )( χ i ( · − τ ) − G hw ∗ χ i ( · − τ )). Using Jensen’s inequality and the elementary identity (150) inthe Appendix we have | χ i − G hw ∗ χ i | ≤ Z G hw ( z ) | χ i ( x ) − χ i ( x − z ) | dz = Z G hw ( z ) χ i ( x )(1 − χ i ( x − z )) dz + Z G hw ( z )(1 − χ i ( x )) χ i ( x − z ) dz = X k = i Z G hw ( z ) χ i ( x ) χ k ( x − z ) dz + X k = i Z G hw ( z ) χ k ( x ) χ i ( x − z ) | dz. (83)Now observe that by testing (42) with G w /K ij (which is bounded, and thus admis-sible), we learn that (84) lim h ↓ √ h Z G hw ( z ) χ i ( x ) χ k ( x − z ) dz = Z G w ( z )( ν ik ( x, t ) · z ) + dz H d − | Σ ik ( t ) ( dx ) dt. Thus, if we divide (83) by √ h and let h ↓
0, using also (70) we obtain α | ∂ t χ i | ≤ lim inf h ↓ | δχ i |√ h ≤ lim sup h ↓ | δχ i |√ h ≤ Cα ω + C H d − ∂ ∗ Ω i ( t ) ( dx ) dt, (85)where C is a constant which depends on γ, β, N , the mobilities and the surface ten-sions. If we divide by α and then let α → | ∂ t χ i | is absolutely continu-ous with respect to H d − | ∂ ∗ Ω i ( t ) ( dx ) dt . In particular, there exists V i ∈ L ( H d − | ∂ ∗ Ω i ( t ) ( dx ) dt )which is the normal velocity of χ i in the sense that ∂ t χ i = V i |∇ χ i | in the sense ofdistributions. The optimal integrability V i ∈ L ( H d − | ∂ ∗ Ω i ( t ) ( dx ) dt ) will be shown inthe second part of the proof. Let us record for later use that with a similar rea-soning we actually obtain that lim sup h | δχ i |√ h is absolutely continuous with respectto H d − | ∂ ∗ Ω i ( t ) ( dx ) dt .Thus in particular inequality (85) holds with ω replaced by itsabsolutely continuous part with respect to H d − | ∂ ∗ Ω i ( t ) ( dx ) dt ; calling this ω aci , it means(86) lim sup h ↓ | δχ i |√ h ≤ Cα ω aci + C H d − | ∂ ∗ Ω i ( t ) ( dx ) dt. Sharp Bound.
Before entering into the proof of the sharp bound, we need to provethe following property. For any i = j we have that, in a distributional sense, thefollowing holds(87) lim h ↓ √ h δχ + i K hij ∗ δχ + j = 0 = lim h ↓ √ h δχ − i K hij ∗ δχ − j . We focus on the first limit, the second one being analogous. The first observationis that the limit(88) λ := lim h ↓ √ h δχ + i K hij ∗ δχ + j E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 27 is a nonnegative bounded measure, which is absolutely continous with respect to H d − | Σ ij ( t ) ( dx ) dt . Indeed, spelling out the z integral and using the fact that δχ + i = χ i (1 − χ i ( − τ )) we obtain1 √ h δχ + i K hij ∗ δχ + j = 1 √ h Z K hij ( z ) δχ + i ( x, t ) ∗ δχ + j ( x − z, t ) dz (89) ≤ √ h Z K hij ( z ) χ i ( x, t ) χ j ( x − z, t ) dz (90)which by (42) in Lemma 4, as h ↓
0, converges to(91) Z K ij ( z )( ν ij ( x, t ) · z ) + H d − | Σ ij ( t ) ( dx ) dt which is absolutely continous with respect to H d − | Σ ij ( t ) ( dx ) dt .Now, given ν ∈ S d − we claim that λ ≤ Z ν · z ≤ K ij ( z )( ν ij · z ) + H d − | Σ ij ( t ) ( dx ) dt + Z ν · z ≥ K ij ( z )( ν ij · z ) − H d − | Σ ij ( t ) ( dx ) dt. (92)To see this, let us denote momentarily the right-hand side of (88) (disintegrated inthe z -variable) as λ h := χ i ( x, t )(1 − χ i )( x, t − τ ) K hij ( z ) χ i ( x − z, t )(1 − χ i )( x − z, t − τ ).Using the fact that 0 ≤ χ i , χ j ≤ P l χ l = 1 we obtain the following inequalities λ h ≤ χ i ( x, t ) K hij ( z ) χ i ( x − z, t ) . (93) λ h ≤ χ j ( x, t − τ ) K hij ( z ) χ i ( x − z, t − τ )+ C X k = i,j K hij ( z ) ( | δχ k | ( x, t ) + | δχ k | ( x − z, t )) . (94)Here C is a constant that does not depend on h . Using inequality (93) on the domain { ν · z ≤ } and inequality (94) on the domain { ν · z ≥ } we obtain λ ≤ lim sup h ↓ √ h Z ν · z ≤ χ i ( x, t ) K hij ( z ) χ i ( x − z, t ) dz + lim sup h ↓ √ h Z ν · z ≥ χ j ( x, t − τ ) K hij ( z ) χ i ( x − z, t − τ ) dz + C X k = i,j lim sup h ↓ (cid:18) √ h Z K hij ( z ) | δχ k | ( x, t ) dz + 1 √ h Z K hij ( z ) | δχ k | ( x − z, t ) dz (cid:19) . Observe that for any 1 ≤ k ≤ N we have(95) lim sup h ↓ √ h Z K hij ( z ) | δχ k | ( x, t ) dz = 1 √ h Z K hij ( z ) | δχ k | ( x − z, t ) dz. This can be seen by showing that(96) lim h ↓ √ h Z K hij ( z ) ( | δχ k | ( x, t ) − | δχ k | ( x − z, t )) dz which can be shown to be true by testing with an admissible test function, andputting the spatial shift z on it. Thus recalling (42) and (86), we obtain that λ ≤ Z ν · z ≤ K ij ( z )( ν ij · z ) + H d − | Σ ij ( t ) ( dx ) dt + Z ν · z ≥ K ij ( z )( ν ij · z ) − H d − | Σ ij ( t ) ( dx ) dt + C X k = i,j α ω ack + H d − | ∂ ∗ Ω k ( t ) ( dx ) dt. (97)Since we already know that λ is absolutely continous with respect to H d − | Σ ij ( t ) ( dx ) dt ,the same bound holds true if we replace the right hand side with its absolutelycontinuous part with respect to H d − | Σ ij ( t ) ( dx ) dt . Observing that for k = i, j by Lemma6 in the Appendix the measures H d − | ∂ ∗ Ω k ( t ) ( dx ) dt and H d − | ∂ ∗ Σ ij ( t ) ( dx ) dt are mutuallysingular , this yields (92).Writing λ = θ ( x, t ) H d − | Σ ij ( t ) ( dx ) dt for some L ( H d − | Σ ij ( t ) ( dx ) dt )-function θ we obtainthat inequality (92) yields E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 29 θ ( x, t ) ≤ Z ν · z ≤ K ij ( z )( ν ij ( x, t ) · z ) + dz Z ν · z ≥ K ij ( z )( ν ij ( x, t ) · z ) − dz (98)for every ν ∈ S d − and H d − | Σ ij ( t ) ( dx ) dt -a.e. ( x, t ) ∈ [0 , d × (0 , T ). By a separabilityargument, we see that the null set on which (98) does not hold can be chosen sothat it is independent of the choice of ν . If we select ν = ν ij ( x, t ) this yields θ ≤ H d − | Σ ij ( t ) ( dx ) dt . Since we already know that λ isnonnegative this gives λ = 0.Before getting the sharp bound, we also need to check that V ij is well defined,i.e. we need to prove that for any i = j we have V i = − V j a.e. with respect to H d − | Σ ij ( t ) ( dx ) dt . To see this, we start by observing that if ξ ∈ C ∞ c ([0 , d × (0 , T )),thanks to the fact that P k = i χ k = 1 − χ i , we get Z ξV i H d − | ∂ ∗ Ω i ( t ) ( dx ) dt = − Z ∂ t ξχ i dxdt = X k = i Z ∂ t ξχ k dxdt = − X k = i Z ξV k H d − | ∂ ∗ Ω k ( t ) ( dx ) dt. (99)Choosing ξ = f ( t ) g ( x ) for some f ∈ C ∞ c ((0 , T )) and g ∈ C ∞ ([0 , d ), by a separabil-ity argument, we obtain that for a.e. t and every g ∈ C ∞ ([0 , d ) Z gV i H d − | ∂ ∗ Ω i ( t ) ( dx ) = − X k = i Z gV k H d − | ∂ ∗ Ω k ( t ) ( dx ) . (100)Pick t such that (100) holds. Let g ∈ C ∞ ([0 , d ) and let ρ m be a partition ofunity obtained by the construction of Section 4 applied to the function χ ( · , t ) on theinterface Σ ij ( t ). Then X m ∈ N Z ρ m gV i H d − | ∂ ∗ Ω i ( t ) ( dx ) = − X m ∈ N X k = i Z ρ m gV k H d − ∂ ∗ Ω k ( t ) ( dx ) . (101)Passing to the limit r ↓ Z gV i H d − | Σ ij ( t ) ( dx ) = − Z gV j H d − ij ( t ) ( dx ) . (102)Since this identity holds for any g ∈ C ∞ ([0 , d ), a density argument gives V i ( x, t ) = − V j ( x, t ) for H d − | Σ ij ( t ) -a.e. x . In other words(103) Z | V i ( x, t ) + V j ( x, t ) |H d − | Σ ij ( t ) ( dx ) = 0 . Integrating in time yields that V i = − V j a.e. with respect to H d − | Σ ij ( t ) ( dx ) dt .We now proceed with the derivation of the sharp lower bound. Define c ij := R K ij ( z ) dz . Then we have c ij ( | δχ i | + | δχ j | ) = c ij ( δχ + i + δχ − j + δχ − i + δχ + j )(104) = 12 (cid:0) δχ + i K hij ∗ (1 − δχ − j ) + (1 − δχ − j ) K hij ∗ δχ + i + δχ − j K hij ∗ (1 − δχ + i )+(1 − δχ + i ) K hij ∗ δχ − j + δχ − i K hij ∗ (1 − δχ + j ) + (1 − δχ + j ) K hij ∗ δχ − i + δχ + j K hij ∗ (1 − δχ − i ) + (1 − δχ − i ) K hij ∗ δχ + j (cid:1) + (cid:0) δχ + i K hij ∗ δχ − j + δχ − j K hij ∗ δχ + i + δχ − i K hij ∗ δχ + j + δχ + j K hij ∗ δχ − i (cid:1) Now we rewrite the terms in the second parenthesis using − ab = a + b − + a − b + − a + b + − a − b − and then adding and subtracting the contributions of the minorityphases we obtain c ij ( | δχ i | + | δχ j | ) ≤ (cid:0) δχ + i K hij ∗ (1 − δχ − j ) + (1 − δχ − j ) K hij ∗ δχ + i + δχ − j K hij ∗ (1 − δχ + i )+(1 − δχ + i ) K hij ∗ δχ − j + δχ − i K hij ∗ (1 − δχ + j ) + (1 − δχ + j ) K hij ∗ δχ − i + δχ + j K hij ∗ (1 − δχ − i ) + (1 − δχ − i ) K hij ∗ δχ + j (cid:1) − X l,p δχ l K hlp ∗ δχ p + δχ + i K hij ∗ δχ + j + δχ − i K hij ∗ δχ − j + δχ + j K hij ∗ δχ + i + δχ − j K hij ∗ δχ − i + X { l,p }6 = { i,j } , { l,p } δχ l K hlp ∗ δχ p . (105)Now the main idea is to split the integral of K ij in the definition of c ij into twoparts. More precisely, by the symmetry (19), for any ν ∈ S d − and any V > E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 31 (106) c ij = 2 Z ≤ ν · z ≤ αV K ij ( z ) dz + 2 Z ν · z>αV K ij ( z ) dz. Substituting into (105) and dividing by √ h we obtain2 Z ≤ ν · z ≤ αV K ij ( z ) dz ( | δχ i | + | δχ j | ) √ h (107) = 12 √ h (cid:18) δχ + i K hij ∗ (1 − δχ − j ) + (1 − δχ − j ) K hij ∗ δχ + i + δχ − j K hij ∗ (1 − δχ + i )+ (1 − δχ + i ) K hij ∗ δχ − j + δχ − i K hij ∗ (1 − δχ + j ) + (1 − δχ + j ) K hij ∗ δχ − i + δχ + j K hij ∗ (1 − δχ − i ) + (1 − δχ − i ) K hij ∗ δχ + j − Z ν · z>αV K ij ( z ) dz ( | δχ i | + | δχ j | ) − X lp δχ l K hlp ∗ δχ p + 2 δχ + i K hij ∗ δχ + j + 2 δχ − i K hij ∗ δχ − j + 2 δχ + j K hij ∗ δχ + i + 2 δχ − j K hij ∗ δχ − i + X ( l,p ) =( i,j ) , ( l,p ) =( j,i ) δχ l K hlp ∗ δχ p (cid:19) . We will be interested in bounding the lim inf of the left hand side. Observe thatthe distributional limit of the last five terms is non-positive. Indeed, the limit of firstfour terms vanish distributionally by property (87), while the last term is boundedfrom above by X ( l,p ) =( i,j ) , ( l,p ) =( j,i ) δχ + l K hlp ∗ δχ + p + δχ − l K hlp ∗ δχ − p , which vanish distributionally by property (87). We thus obtain that the lim inf ofthe left hand side of (107) is bounded from above by lim inf h ↓ √ h (cid:18) δχ + i K hij ∗ (1 − δχ − j ) + (1 − δχ − j ) K hij ∗ δχ + i + δχ − j K hij ∗ (1 − δχ + i )+ (1 − δχ + i ) K hij ∗ δχ − j + δχ − i K hij ∗ (1 − δχ + j ) + (1 − δχ + j ) K hij ∗ δχ − i + δχ + j K hij ∗ (1 − δχ − i ) + (1 − δχ − i ) K hij ∗ δχ + j − Z ν · z>αV K ij ( z ) dz ( | δχ i | + | δχ j | ) − X lp δχ l K hlp ∗ δχ p (cid:19) . (108)For the last term we use the sharp bound (69), relating this term to our dissipationmeasure ω . We would like to get a good bound for the other terms. This cannot bedone naively as before, since we want the bound to be sharp. We claim thatlim sup h ↓ √ h (cid:0) δχ + i K hij ∗ (1 − δχ − j ) + (1 − δχ − j ) K hij ∗ δχ + i δχ − j K hij ∗ (1 − δχ + i )+(1 − δχ + i ) K hij ∗ δχ − j + δχ − i K hij ∗ (1 − δχ + j ) + (1 − δχ + j ) K hij ∗ δχ − i + δχ + j K hij ∗ (1 − δχ − i ) + (1 − δχ − i ) K hij ∗ δχ + j − Z ν · z>αV K ij ( z ) dz ( | δχ i | + | δχ j | ) (cid:19) (109) ≤ Z ≤ ν · z ≤ αV K ij ( z ) | ν ij ( x ) · z | dz H d − | Σ ij ( t ) ( dx ) dt + C X k = i,j ( α ω ack + H d − | ∂ ∗ Ω k ( t ) ( dx ) dt. Here C is a constant that depends on γ, β, A , B , but not on h . Assume for themoment that (109) is true and let us conclude the argument in this case. Using(109) and (69) we obtain2 lim inf h ↓ Z ≤ ν · z ≤ αV K ij ( z ) dz ( | δχ i | + | δχ j | ) √ h (110) ≤ α ω + 4 Z ≤ ν · z ≤ αV K ij ( z ) | ν ij ( x ) · z | dz H d − | Σ ij ( t ) ( dx ) dt + C X k = i,j ( α ω ack + H d − | ∂ ∗ Ω k ( t ) ( dx ) dt E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 33 in the sense of distributions on [0 , d × (0 , T ). Observe also that the left handside of (110) is an upper bound for R ≤ ν · z ≤ αV K ij ( z ) dz ( | ∂ t χ i | + | ∂ t χ j | ), thus theinequality still holds true if the left hand side is replaced by this term. Rememberthat ω ack is absolutely continuous with respect to H d − | ∂ ∗ Ω k ( t ) ( dx ) dt , thus there existfunctions W k ∈ L ( H d − | ∂ ∗ Ω k ( t ) ( dx ) dt ) such that ω ack = W k ( x, t ) H d − | ∂ ∗ Ω k ( t ) ( dx ) dt . We nowdisintegrate the measure ω , i.e. we find a Borel family ω t , t ∈ (0 , T ), of positivemeasures on [0 , d such that ω = ω t ⊗ dt . Having said this, it is not hard to see that(110) holds in a disintegrated version, i.e. we have for Lebesgue a.e. t ∈ (0 , T )2 Z ≤ ν · z ≤ αV K ij ( z ) dz ( | V i ( x, t ) |H d − | ∂ ∗ Ω i ( t ) ( dx ) + | V j ( x, t ) |H d − | ∂ ∗ Ω j ( t ) ( dx ))(111) ≤ α ω t + 4 Z ≤ ν · z ≤ αV K ij ( z ) | ν ij ( x ) · z | dz H d − | Σ ij ( t ) ( dx )+ C X k = i,j ( α W k ( x, t ) + 1) H d − | ∂ ∗ Ω k ( t ) ( dx ) . Here ν ∈ S d − and V ∈ (0 , ∞ ) are arbitrary: indeed even if the set of points intime for which (111) holds is a priori dependent on ν and V , a standard separabilityargument allows us to conclude that we can get rid of this dependence.Fix a point t in time such that (111) holds. In what follows, we drop the timevariable t which is fixed, so for example V i ( x ) = V i ( x, t ), Σ ij = Σ ij ( t ) and so on.Fix ξ ∈ C ([0 , d ), observe that by definition of V ij and by using the fact thatΣ ij ⊂ ∂ ∗ Ω i ∩ ∂ ∗ Ω j we have4 α Z ≤ ν · z ≤ αV K ij ( z ) dz Z [0 , d ξ ( x ) | V ij ( x ) |H d − | Σ ij ( dx )(112) ≤ α Z [0 , d ξ ( x ) ω t ( dx ) + 4 Z [0 , d Z ≤ ν · z ≤ αV K ij ( z ) | ν ij ( x ) · z | dzξ ( x ) H d − | Σ ij ( t ) ( dx )+ C X k = i,j Z [0 , d ξ ( x )( α W k ( x, t ) + 1) H d − | ∂ ∗ Ω k ( t ) ( dx ) . Let us relabel ν , V and ξ to make clear that they may depend on the pair i, j .Thus ν ij ∈ S d − , V ij ∈ (0 , ∞ ) and ξ ij ∈ C ([0 , d ) are arbitrary, and it holds α Z ≤ ν ij · z ≤ αV ij K ij ( z ) dz Z [0 , d ξ ij ( x ) | V ij ( x ) |H d − | Σ ij ( dx )(113) ≤ α Z [0 , d ξ ij ( x ) ω t ( dx ) + 4 Z [0 , d Z ≤ ν ij · z ≤ αV ij K ij ( z ) | ν ij ( x ) · z | dzξ ij ( x ) H d − | Σ ij ( t ) ( dx )+ C X k = i,j Z [0 , d ξ ij ( x )( α W k ( x, t ) + 1) H d − | ∂ ∗ Ω k ( t ) ( dx ) . Let { ρ m } be a partition of unity obtained using the construction of Section 4applied to the function χ ( · , t ) on the inferface Σ ij ( t ). Use the above inequality with ξ ij replaced by ρ m ξ ij and sum over m and i, j to get X i 0. To see this, it is clear that we can concentrate on ξ ij = w ij B ij ,where B ij ⊂ [0 , d are Borel and w ij ≥ 0. Observe that by the dominated conver-gence theorem, the family(117) F := ( B = Y i To prove (109) we proceed in several steps.First of all, we claim that the first eight terms may be substituted by2 Z ν · z ≥ K hij ( z ) (cid:0) | δχ + i − δχ − j ( − z ) | + | δχ + i ( − z ) − δχ − j | (125) | δχ − i − δχ + j ( − z ) | + | δχ − i ( − z ) − δχ + j | (cid:1) dz. To show this, observe that we may replace the implicit z -integrals in the convo-lution in the first eight terms by twice the integrals over the half space { ν · z ≥ } instead of R d . This is clearly true once we observe thatlim h ↓ √ h (cid:18) δχ + i Z ν · z ≥ K hij ( z )(1 − δχ − j ( · − z )) dz +(1 − δχ − j ) Z ν · z ≥ K hij ( z ) δχ + i ( · − z ) dz (cid:19) (126) = lim h ↓ √ h (cid:18) δχ + i Z ν · z ≤ K hij ( z )(1 − δχ − j ( · − z )) dz +(1 − δχ − j ) Z ν · z ≤ K hij ( z ) δχ + i ( · − z ) dz (cid:19) and that similar identities hold exchanging the roles of i, j and + , − respectively.That (126) holds is not difficult to show. Indeed taking into account the fact thatthe kernel is even, the argument of the second limit is just a spatial shift of z of thefirst one. The spatial shift may be put onto the test function, and thanks to thescaling of the kernel one can get the claim. We may thus substitute the first eightterms of the left hand side of (109) with twice the same terms with the integrationwith respect to z on the half space { ν · z ≥ } . If we rely again on the fact that δχ + i ∈ { , } , by identity (150) in the Appendix we obtain (125), as claimed.Now we need two inequalities for the integrand. First note that the integrand is asecond-order finite difference, we claim that | δχ + i − δχ − j ( · − z ) | + | δχ + i ( · − z ) − δχ − j | + | δχ − i − δχ + j ( · − z ) | + | δχ − i ( · − z ) − δχ + j | (127) ≤ | δχ + i − δχ + i ( · − z ) | + | δχ − i − δχ − i ( · − z ) | + | δχ + j − δχ + j ( · − z ) | + | δχ − j − δχ − j ( · − z ) | +4 P k = i,j ( | δχ k | + δχ k ( · − z ) | ) . | δχ i | + | δχ i ( · − z ) | + | δχ j | + | δχ j ( · − z ) | . The second one follows from the triangle inequality. To show the first one, observethat | δχ + i − δχ − j ( · − z ) | =(1 − δχ + i ) δχ − j ( · − z ) + δχ + i (1 − δχ − j ( · − z ))(128) ≤ (1 − δχ + i ) δχ + i ( · − z ) + X k = i,j | δχ k ( · − z ) | + δχ + j (1 − δχ + j ( · − z ))+ X k = i,j | δχ k | and that similarly | δχ + i ( · − z ) − δχ − j | =(1 − δχ i ( · − z ) + ) δχ − j + δχ i ( · − z ) + (1 − δχ j )(129) E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 39 ≤ (1 − δχ i ( · − z ) + ) δχ − i + X k = i,j | δχ k | + δχ j ( · − z ) + (1 − δχ − j )+ X k = i,j | δχ k ( · − z ) | . Summing up the two inequalities we get | δχ + i − δχ − j ( · − z ) | + | δχ + i ( · − z ) − δχ − j | ≤ (130) ≤ | δχ + i − δχ + i ( · − z ) | + | δχ − j − δχ − j ( · − z ) | + 2 X k = i,j | δχ k | + | δχ k ( · − z ) | . Similar bounds hold for the remaining terms in (127).We now split the integral (125) into the domains of integration { ≤ ν · z ≤ αV } and { ν · z > αV } . On the first one we use the first inequality in (127) for theintegrand. Recalling identity (150) and inequality (151) in the Appendix we obtain,and using the fact that P k χ k = 12 Z ≤ ν · z ≤ αV K hij ( z ) (cid:0) | δχ + i − δχ − j ( · − z ) | + | δχ + i ( · − z ) − δχ − j | + | δχ − i − δχ + j ( · − z ) | + | δχ − i ( · − z ) − δχ + j | (cid:1) dz ≤ Z ≤ ν · z ≤ αV K hij ( z ) ( | χ i − χ i ( · − z ) | + | χ i ( − τ ) − χ i ( · − τ, · − z ) || χ j − χ j ( · − z ) | + | χ j ( · − τ ) − χ j ( · − τ, · − z ) | +8 X k = i,j | δχ k | + | δχ k ( · − z ) | ! dz ≤ Z ≤ ν · z ≤ αV K hij ( z ) χ i χ j ( · − z ) + χ i ( · − z ) χ j + X k = i,j χ i χ k ( · − z ) + χ i ( · − z ) χ k χ i ( · − τ ) χ j ( · − τ, · − z ) + χ i ( · − τ, · − z ) χ j ( · − τ )+ X k = i,j χ i ( · − τ ) χ k ( · − τ, · − z ) + χ i ( · − τ, · − z ) χ k ( · − τ ) χ j χ i ( · − z ) + χ j ( · − z ) χ i + X k = i,j χ j χ k ( · − z ) + χ j ( · − z ) χ k χ j ( · − τ ) χ i ( · − τ, · − z ) + χ j ( · − τ, · − z ) χ i ( · − τ )+ X k = i,j χ j ( · − τ ) χ k ( · − τ, · − z ) + χ j ( · − τ, · − z ) χ k ( · − τ )+8 X k = i,j | δχ k | + | δχ k ( · − z ) | ! dz. (131)On the set { ν · z > αV } we use the second inequality in (127), obtaining2 Z ν · z>αV K hij ( z ) (cid:0) | δχ + i − δχ − j ( · − z ) | + | δχ + i ( · − z ) − δχ − j | + | δχ − i − δχ + j ( · − z ) | + | δχ − i ( · − z ) − δχ + j | (cid:1) dz (132) ≤ Z ν · z>αV K hij ( z )( | δχ i | + | δχ i ( · − z ) | + | δχ j | + | δχ j ( · − z ) | ) dz. We now observe that for any 1 ≤ k ≤ N we have, as we already observed in (96)(133) lim h ↓ √ h Z ≤ ν · z ≤ αV K hij ( z )( | δχ k ( · − z ) | − | δχ k | ) dz = 0 , thus in particular lim sup h ↓ √ h Z ≤ ν · z ≤ αV K hij ( z ) | δχ k ( · − z ) | dz (134) = lim sup h ↓ √ h Z ≤ ν · z ≤ αV K hij ( z ) | δχ k | dz. By putting the time shift τ on the test function if is easy to check that the distri-butional limit of the terms of (131) which involve the shift τ have the same limit asthe corresponding terms without the time shift. Thus recalling (86) and relying on E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 41 (133) and (42) we obtain that inserting (131) and (132) into (125), the left hand sideof (109) is bounded by8 Z ≤ ν · z ≤ αV K ij ( z )(( ν ij ( x, t ) · z ) + + ( ν ij ( x, t ) · z ) − ) H d − | Σ ij ( t ) ( dx ) dt + C X k = i,j Z ≤ ν · z ≤ αV K ik ( z )(( ν ik ( x, t ) · z ) + + ( ν ik ( x, t ) · z ) − ) H d − | Σ ik ( t ) ( dx ) dt + C X k = i,j ( α ω ack + H d − | ∂ ∗ Ω k ( t ) ( dx ) dt ) , which clearly gives the claim once we realize that Z ≤ ν · z ≤ αV K ik ( z )(( ν ik ( x ) · z ) + + ( ν ik ( x ) · z ) − ) H d − | Σ ik ( t ) ( dx ) dt ≤ Z R d K ik ( z ) | z | dz ≤ C. (cid:3) Proof of Proposition 2. The proof is along the same lines as Proposition 2 in [19],where the claim is analized in the case of two phases. For the convenience of thereader, we outline the strategy of the full proof, providing details only for the requiredchanges. The proof is split into several steps. step 1 . The first observation is that for any h > 0, any admissible u ∈ M and anysmooth vector field ξ we have the following lower bound for the metric slope, cf. (33)12 | ∂E h | ( u ) ≥ δE h ( u ) • ξ − 12 ( δd h ( · , u ) • ξ ) . Here δ denotes the first variation, which is computed considering the curve s → u s of configurations which solve the transport equations(135) ( ∂ s u si + ξ · ∇ u si = 0 ,u si ( · , 0) = u i ( · ) , and by setting(136) δE h ( u ) • ξ := dds | s =0 E h ( u s ) and δd h ( · , u ) • ξ := dds | s =0 d ( u, u s ) . step 2 . The second observation is a representation formula for δE h ( u ) • ξ . Namely δE h ( u ) • ξ = X i,j √ h (cid:18)Z ∇ · ξu i K hij ∗ u j dx + Z ∇ · ξu j K hij ∗ u i dxdt + Z [ ξ, ∇ K hij ∗ ]( u j ) u i dx (cid:19) . (137)Here [ ξ, ∇ K hij ∗ ] denotes the commutator obtained taking the convolution with ∇ K hij and multiplying by ξ . To check this formula one starts by assuming u to be smoothand then an approximation argument gives the result for a general u ∈ M . step 3 . Representation for δd h ( · , u ) • ξ . One checks that12 ( δd h ( · , u ) • ξ ) = √ h X i,j (cid:18)Z u i ξ · ∇ K hij ∗ ( ξu j ) dx + Z u j ξ · ∇ K hij ∗ ( ξu i ) dx + Z u i ∇ · ξ ∇ K hij ∗ ( ξu j ) dx + + Z u j ∇ · ξ ∇ K hij ∗ ( ξu i ) dx − Z u i ∇ · ξK hij ∗ ( u j ∇ · ξ ) dx − Z u j ∇ · ξK hij ∗ ( u i ∇ · ξ ) dx − Z ξu i ∇ K hij ∗ ( u j ∇ · ξ ) dx − Z ξu j ∇ K hij ∗ ( u i ∇ · ξ ) dx (cid:19) . Once again this formula can be easily checked when u is smooth, an approximationargument then gives the extension to the case u ∈ M . step 4 . Passage to the limit in δE h . We claim that(138) lim h ↓ Z T δE h ( u h ( t )) • ξdt = X i,j σ ij Z ( ∇ · ξ − ν ij · ∇ ξν ij ) H d − | Σ ij ( t ) ( dx ) dt. The proof is very similar to the two phases case, and relies on the weak convergence(42). Firstly, testing (42) with ∇ · ξ we getlim h ↓ X i,j √ h Z (cid:0) ∇ · ξu hi K hij ∗ u hj + ∇ · ξu hj K hij ∗ u hi (cid:1) dxdt = X i,j σ ij Z ∇ · ξ H d − | Σ ij ( t ) ( dx ) dt. For the term involving the commutator, one checks that E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 43 lim h ↓ (cid:18)Z [ ξ, ∇ K hij ∗ ]( u hj ) u hi dxdt − Z ∇ ξz · ∇ K hij ( z ) u hj ( x − z, t ) u hi ( x, t ) dzdxdt (cid:19) = 0 . With this in place, we observe that Z ∇ ξz · ∇ K hij ( z ) u hj ( x − z, t ) u hi ( x, t ) dzdxdt = Z ∇ ξ ( x, t ) z · ∇ K ij ( z )( ν ij ( x, t ) · z ) + H d − | Σ ij ( t ) ( dx ) dt which can be seen by testing (42) with ∇ ξz ·∇ K ij ( z ) K ij ( z ) which is of polynomial growth in z . To conclude (138) we just need to show that for any symmetric matrix A ∈ R d × d and any unit vector ν we have Z Az · ∇ K ij ( z )( ν · z ) + dz = − σ ij (tr A + ν · Aν ) . Using the definition of the kernel K ij it suffices to show that Z Az · ∇ G w ( z )( ν · z ) + dz = − √ w √ π (tr A + ν · Aν ) w ∈ { γ, β } . step 5 . Passage to the limit in δd h ( · , u ) ξ . We claim thatlim h ↓ (cid:0) δd h ( · , u h ) • ξ (cid:1) = X i,j µ ij Z ( ξ · ν ij ) H d − | Σ ij ( t ) ( dx ) dt. (139)To prove this, we observe that the terms which do not involve the Hessian ∇ K hij are all O ( √ h ). For example, to prove that(140) √ h Z u hi ∇ · ξ ∇ K hij ∗ ( ξu hj ) dxdt = O ( √ h ) , spell out the integral in the convolution, use the fact that ∇ K hij = √ h d +1 ∇ K ij ( z √ h ),use the fact that ∇ ξ ( x, t ) ξ ( x − √ hz, t ) is bounded and test (42) with ∇ K ij /K ij .The other terms can be treated similarly. For the terms involving the Hessian of thekernel, we split the claim intolim h ↓ √ h Z u hi ( ξ · ∇ K hij ∗ u j ) ξdxdt = 12 µ ij Z ( ξ · ν ij ( x, t )) H d − | Σ ij ( t ) ( dx ) dt, (141) √ h Z u hi ξ · [ ξ, ∇ K hij ∗ ]( u hj ) dxdt = O ( √ h ) . (142)The proof of (142) is similar to the argument for (140). To prove identity (141)observe that by spelling out the z -integral, a change of variable and by testing (42)with ξ ( x,t ) ·∇ K ij ( z ) ξ ( x,t ) K ij ( z ) we obtainlim h ↓ √ h Z u hi ( ξ · ∇ K hij ∗ u j ) ξdxdt = Z ξ · ∇ K ij ( z ) ξ ( ν ij ( x, t ) · z ) + H d − | Σ ij ( t ) ( dx ) dt. Now identity (139) follows from the following formula: for any two vectors ξ ∈ R d and ν ∈ S d − we have(143) Z ξ · ∇ K ij ( z ) ξ ( ν · z ) + dz = 12 µ ij ( ξ · ν ) . To check (143), by relying on the definition of the kernels, we just need to showthat for w ∈ { γ, β } Z ξ · ∇ G w ( x ) ξ ( ν · z ) + dz = 12 √ πw ( ξ · ν ) . Since the kernel is isotropic, we can reduce to the case ξ = e , thus we need to prove Z ∂ G w ( x )( ν · z ) + dz = 12 √ πw ν . This can be done after two integration by parts and observing that Z ν · z =0 G w ( z ) dz = 12 √ πw . conclusion . By step 1 we have12 Z T | ∂E h | ( u h ) dt ≥ Z T δE h ( u h ) • ξdt − Z T (cid:0) δd h ( · , u h ) • ξ (cid:1) dt. Taking the liminf on the left hand side, using step 4 and step 5 we get that forany smooth vector field ξ E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 45 lim inf h ↓ Z T | ∂E h | ( u h ) dt ≥ X i,j (cid:20) σ ij Z ( ∇ · ξ − ν ij · ∇ ξν ij ) H d − | Σ ij ( t ) ( dx ) dt − µ ij Z ( ξ · ν ij ) H d − | Σ ij ( t ) ( dx ) dt (cid:21) . Since the left hand side is bounded, the Riesz representation theorem for L yieldsfunctions H ij ∈ L ( H d − | Σ ij ( t ) ( dx ) dt ) such that X i,j σ ij Z ( ∇ · ξ − ν ij · ∇ ξν ij ) H d − | Σ ij ( t ) ( dx ) dt = − X i,j σ ij Z H ij ν ij · ξ H d − | Σ ij ( t ) ( dx ) dt and such that for any ξ ∈ L ( H d − | S i,j Σ ij ( t ) ( dx ) dt )lim inf h ↓ Z T | ∂E h | ( u h ) dt ≥ X i,j (cid:18) − σ ij Z H ij ν ij · ξ H d − | Σ ij ( t ) ( dx ) dt − µ ij Z ( ξ · ν ij ) H d − | Σ ij ( t ) ( dx ) dt (cid:19) . Since the integration measures are mutually singular we can test with ξ ∈ L ( H d − | S i,j Σ ij ( t ) ( dx ) dt )such that ξ | Σ ij ( t ) = − µ ij σ ij H ij ν ij . This yieldslim inf h ↓ Z T | ∂E h | ( u h ) dt ≥ X i,j σ ij µ ij Z H ij H d − | Σ ij ( t ) ( dx ) dt. (cid:3) Appendix Proof of Lemma 5. Before giving the proof of this result, we need a simpletechnical lemma. Lemma 6. Fix ≤ l = p ≤ N . Then for any ≤ i = j ≤ N such that { i, j } 6 = { l, p } the interfaces Σ ij and Σ lp are disjoint. In particular for H d − -a.e. x ∈ Σ lp we havethat (144) lim r → H d − (Σ ij ∩ B ( x, r )) = 0 Proof. We first show that the interfaces Σ ij and Σ lp are disjoint. This follows im-mediately once we recall that every point in the reduced boundary of a set of finiteperimeter has density 1 / i = l, p .Thus if y ∈ Σ lp we have1 ≥ lim sup r | (Ω l ∪ Ω p ∪ Ω i ) ∩ B ( y, r ) | ω d r d = lim r | Ω l ∩ B ( y, r ) | ω d r d + lim r | Ω p ∩ B ( y, r ) | ω d r d + lim sup r | Ω i ∩ B ( y, r ) | ω d r d = 1 + lim sup r | Ω i ∩ B ( y, r ) | ω d r d (145)which says that y has density zero in Ω i .The fact that (144) holds is now a consequence of the geneneral factlim sup r ↓ H d − (Σ ij ∩ B ( x, r )) ω d − r d − = 0for H d − -a.e. x ∈ Σ cij . (cid:3) Proof of Lemma 5. The argument for (1) can be found in [17] in the case of twophases and without localization, i.e. with η = 1 and N = 2. For the sake of com-pleteness, we provide the proof in our case. Upon splitting into the negative andpositive part, we may assume η ≥ 0. Clearly the only nonzero terms in the sum arethose for wich B rm ∩ Σ ij = ∅ . Fix such a ball: by definition there exists y ∈ r Z d suchthat B rm = B ( y, r √ d ). If x ∈ Σ ij ∩ B rm then we have that B ( x, r √ d ) ⊂ B ( y, r √ d ),and by definition of E r this yields H d − ( B ( x, r √ d ) ∩ Σ ij ) ≤ ω d − d (4 r ) d − √ d d − = ω d − r ) d − √ d d − . Thus x belongs to the set of points in Σ ij ∩ B rm such that(146) H d − ( B ( x, r √ d ) ∩ Σ ij ) ω d − (2 r √ d ) d − ≤ . By De Giorgi’s structure theorem the approximate tangent plane exists at everypoint x ∈ Σ ij , thus (146) cannot hold when r is small enough: moreover every point x ∈ Σ ij is contained in at most c (2 , d ) balls, this means that(147) X m (cid:26) z ∈ B rm ∩ Σ ij : H d − B ( x, r √ d ) ∩ Σ ij ) ωd − r √ d ) d − ≤ (cid:27) ( x ) η ( x ) ≤ c (2 , d ) η ( x ) E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 47 and that the left hand side of (147) converges to zero pointwise. By the dominatedconvergence theorem we get our claim.Proof of (2). Upon splitting into the negative and positive part, we may assume η ≥ 0. Given a point x ∈ Σ lp , if y ∈ r Z d is such that x ∈ B ( y, r √ d ), then B ( y, r √ d ) ⊂ B ( x, r √ d ). Thus for any 1 ≤ i < j ≤ N with ( i, j ) = ( l, p ) we have H d − ( B ( y, r √ d ) ∩ Σ ij ) ≤ H d − ( B ( x, r √ d ) ∩ Σ ij ) ≤ ω d − d (4 r ) d − √ d d − provided r is small enough, this follows from Lemma 6. Since F r covers R d we obtainthat x ∈ [ m ∈ N B rm for all r small enough. In other wordslim r ↓ X m ρ m ( x ) η ( x ) = η ( x )pointwise on Σ lp , and the argument of the limit on the right hand side is dominatedby η . Thus we may once again appeal to the dominated convergence theorem andconclude the proof. (cid:3) Consistency and Monotonicity. The following results are essetially con-tained in [8] and [17], indeed the proofs may be adapted because we are assumingthat a ij and b ij satisfy the triangle inequality. Lemma 7. Let χ ∈ L ((0 , T ) , BV ([0 , d ) N ) such that χ ( · , t ) ∈ A for a.e. t . Then lim h ↓ Z T E h ( χ ) dt = Z T E ( χ ) dt. Even more is true: for any g ∈ C ∞ ([0 , d ) and any pair ≤ i = j ≤ N we have lim h ↓ √ h Z T Z g ( x )( χ i ( x, t ) K hij ∗ χ j ( x, t ) + χ j ( x, t ) K hij ∗ χ j ( x, t )) dxdt = Z g ( x ) K ij ( z ) | ν ij · z | dzdxdt. Lemma 8. For any < h ≤ h we have E h ( u ) ≥ (cid:18) √ h √ h + √ h (cid:19) d +1 E h ( u ) . Improved convergence of the energies. The following Lemma is an im-provement of the convergence of the energies, the proof of this result is contained,with minor modifications, in the paper [17], Corollary 3.7. Lemma 9. Let u h be a sequence of [0 , -valued functions such that u h → χ in L ([0 , d × (0 , T )) and (148) lim h ↓ Z T E h ( u h ( t )) dt = Z T E ( χ ( t )) dt. Then we have that lim h ↓ √ h Z G hγ ( z ) | f γh ( z ) − f γ ( z ) | dz = 0 , (149) lim h ↓ √ h Z G hβ ( z ) | f βh ( z ) − f β ( z ) | dz = 0 . Where we set f γh ( z ) = X i,j a ij Z u hi ( x, t ) u hj ( x − z, t ) dxdt, f γ ( z ) = X i,j a ij Z χ i ( x, t ) χ j ( x − z, t ) dxdt,f βh ( z ) = X i,j b ij Z u hi ( x, t ) u hj ( x − z, t ) dxdt, f β ( z ) = X i,j b ij Z χ i ( x, t ) χ j ( x − z, t ) dxdt. Some inequalities. Here we gather some elementary inequalities which areused frequently. Lemma 10. Let a, b, a ′ , b ′ ∈ { , } , then the following inequalities hold. | a − b | = a (1 − b ) + b (1 − a )(150) | ( a − a ′ ) + − ( b − b ′ ) + | + | ( a − a ′ ) − − ( b − b ′ ) − |≤ | a − b | + | a ′ − b ′ | (151) Proof. The first identity follows by expanding | a − b | = | a − b | . The second one isproved in [19]. For the sake of completeness, we reproduce the proof here. There aretwo cases. In the first one we have ( a − a ′ )( b − b ′ ) ≥ E GIORGI’S INEQUALITY FOR THE THRESHOLDING SCHEME 49 replacing ( a, a ′ , b, b ′ ) with ( − a, − a ′ , − b, − b ′ ) that ( a − a ′ ) and ( b − b ′ ) are non-negative.Then (151) reduces to | ( a − a ′ ) − ( b − b ′ ) | ≤ | a − b | + | a ′ − b ′ | . The second case is given by ( a − a ′ )( b − b ′ ) ≤ 0. By an argument as before we mayassume ( a − a ′ ) ≥ ≥ ( b − b ′ ), thus (151) reduces to( a − a ′ ) + ( b − b ′ ) ≤ | a − b | + | a ′ − b ′ | . (cid:3) Lemma 11. There exists a constant C > depending only on N, A , B such that forany v ∈ M (152) Z | v − K h ∗ v | dx ≤ C p h E h ( v ) for all h ≥ h. Proof. The proof of (152) is aontained in the proof of Lemma 3 in [19] for the twophases case when K h is the scaled version of the Gaussian with variance 1. 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Springer,Singapore, 2019.(Tim Laux) Institut f¨ur angewandte Mathematik, Universit¨at Bonn, EndenicherAllee 60, 53115 Bonn, Germany Email address : [email protected] (Jona Lelmi) Institut f¨ur angewandte Mathematik, Universit¨at Bonn, EndenicherAllee 60, 53115 Bonn, Germany Email address ::