METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES IN TWO DIMENSIONS: A Γ-CONVERGENCE APPROACH

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1 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES IN TWO DIMENSIONS: A Γ-CONVERGENCE APPROACH R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE Abstract. This paper aims at building a variational approach to the dynamics of discrete topological singularities in two dimensions, based on Γ- convergence. We consider discrete systems, described by scalar functions defined on a square lattice and governed by periodic interaction potentials. Our main motivation comes from XY spin systems, described by the phase parameter, and screw dislocations, described by the displacement function. For these systems, we introduce a discrete notion of vorticity. As the lattice spacing tends to zero we derive the first order Γ-limit of the free energy which is referred to as renormalized energy and describes the interaction of vortices. As a byproduct of this analysis, we show that such systems exhibit increasingly many metastable configurations of singularities. Therefore, we propose a variational approach to depinning and dynamics of discrete vortices, based on minimizing movements. We show that, letting first the lattice spacing and then the time step of the minimizing movements tend to zero, the vortices move according with the gradient flow of the renormalized energy, as in the continuous Ginzburg-Landau framework. Contents. Introduction. The discrete model for topological singularities 7.. Discrete functions and discrete topological singularities 8.. The discrete energy 9 3. Localized lower bounds The zero-order Γ-convergence Lower bound on annuli 3.3. Ball Construction 3.4. Proof of Theorem The renormalized energy and the first order Γ-convergence Revisiting the analysis of Bethuel-Brezis-Hélein The main Γ-convergence result The proof of Theorem Γ-convergence analysis in the L topology 5. Analysis of local minimizers Antipodal configurations and energy barriers Metastable configurations and pinning 6 6. Discrete gradient flow of F ε with flat dissipation Flat discrete gradient flow of W Flat discrete gradient flow of F ε 33

2 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE 7. Discrete gradient flow of F ε with L dissipation L discrete gradient flow of W L discrete gradient flow of F ε Conclusions 50 Appendix A. Product-Estimate 5 References 53. Introduction Phase transitions mediated by the formation of topological singularities characterize many physical phenomena such as superconductivity, superfluidity and plasticity. For its central role in Materials Science, this subject has attracted much attention in the last decades ([9], [3], [3], see also [43]), and has brought new interest on fascinating research fields in the mathematical community, as in the theory of harmonic maps on manifolds ([4], [5], [0]). In particular, new variational methods have been developed to describe and predict the relevant phenomena, such as the formation of topological singularities and the corresponding concentration of energy. Two paradigmatic examples of the appearance of topological singularities are given by screw dislocations in crystals and vortices in superconductors. We now introduce two basic discrete models to describe these phenomena. Given an open set Ω R, consider the square lattice εz Ω, representing the reference configuration of our physical system. In the case of screw dislocations we consider the elastic energy defined on scalar functions u : εz Ω R given by (.) SD ε (u) := dist (u(i) u(j), Z). i,j εz Ω, i j =ε Here ε represents the lattice spacing of a cubic lattice casted in a cylindrical crystal, εz Ω is a reference planar section of the crystal, and u represents the vertical displacement (scaled by /ε). The periodicity of the energy is consistent with the fact that plastic deformations, corresponding to integer jumps of u, do not store elastic energy, according with Nabarro Peierls and Frenkel Kontorova theories [5]. Potentials as in (.) are commonly used in models for dislocations (see e.g. [7], [7], [], [36]; see also [8] for more general discrete lattice energies accounting for defects). A celebrated discrete model which allows to describe the formation of topological singularities, as vortices in superconductors, is the so-called XY spin model. Here, the order parameter is a vectorial spin field v : εz Ω S and the corresponding energy is given by XY ε (v) := v(i) v(j). i,j εz Ω i j =ε Notice that XY ε (v) can be written in terms of a representative of the phase of v, defined as a scalar field u such that v = e πiu. In this respect, both models can be regarded as specific examples of scalar systems governed by periodic potentials f acting on first neighbors, whose energy is of the type F ε (u) := f(u(i) u(j)). i,j εz Ω, i j =ε

3 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 3 How do dislocations or vortices enter in this description? Loosely speaking, they are defined through a discrete notion of topological degree of the field v = e πiu ; they are point singularities, and can be identified by the discrete vorticity measure µ(u). This is a finite sum of Dirac masses centered in the squares of the lattice, and with multiplicities equal to either + or. This notion in the case of dislocations corresponds to the discrete circulation of the plastic strain, and µ(u) represents the Nye dislocation density. This paper aims at studying the statics and the dynamics of such topological singularities, by variational principles. The first step is the asymptotic analysis by Γ-convergence of the discrete energies F ε, as ε 0. This analysis relies on the powerful machinery developed in the recent past for the analysis of Ginzburg-Landau functionals, which can be somehow considered the continuous counterpart of the energies F ε. We recall that, for a given ε > 0, the Ginzburg-Landau energy GL ε : H (Ω; R ) R is defined by (.) GL ε (w) = Ω w dx + ε Ω ( w ) dx. Starting from the pioneering book [0], the variational analysis as ε 0 of GL ε has been the subject of a vast literature. The analysis in [0] shows that, as ε tends to zero, vortex-like singularities appear by energy minimization (induced for instance by the boundary conditions), and each singularity carries a quantum of energy of order log ε. Removing this leading term from the energy, a finite quantity remains, called renormalized energy, depending on the positions of the singularities. This asymptotic analysis has been also developed through the solid formalism of Γ-convergence ([9], [30], [38], [40], [3]). It turns out that the relevant object to deal with is the distributional Jacobian Jw, which, in the continuous setting, plays the role of the discrete vorticity measure. A remarkable fact is that these results also contain a compactness statement. Indeed, for sequences with bounded energy the vorticity measure is not in general bounded in mass; this is due to the fact that many dipoles are compatible with a logarithmic energy bound. Therefore, the compactness of the vorticity measures fails in the usual sense of weak star convergence. Nevertheless, compactness holds in the flat topology, i.e., in the dual of Lipschitz continuous functions with compact support. Recently, part of this Γ-convergence analysis has been exported to two-dimensional discrete systems. In [35], [], [] it has been proved that the functionals log ε F ε Γ-converge to π µ(ω), where µ is the limiting vorticity measure and is given by a finite sum of Dirac masses. This Γ-limit is not affected by the position of the singularities and hence does not account for their interaction, which is an essential ingredient in order to study the dynamics. In this paper, we make a further step in this direction, deriving the renormalized energy for our discrete systems by Γ- convergence, using the notion of Γ-convergence expansion introduced by [7] (see also [4]). Precisely, in Theorem 4. we prove that, given M N, the functionals F ε (u) Mπ log ε Γ-converge to W(µ) + Mγ, where µ is a sum of M singularities x i with degrees d i = ±. Here W is the renormalized energy as in the Ginzburg- Landau setting, defined by W(µ) := π i j d i d j log x i x j π i d i R 0 (x i ),

4 4 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE where R 0 is a suitable harmonic function (see (4.)), and γ can be viewed as a core energy, depending on the specific discrete interaction energy (see (4.6)). An intermediate step to prove Theorem 4. is Theorem 3. (ii), which establishes a localized lower bound of the energy around the limiting vortices. This result is obtained using a tool introduced by Sandier [37] and Jerrard [9] for the functionals GL ε, referred to as ball construction; it consists in providing suitable pairwise disjoint annuli, where much of the energy is stored, and estimating from below the energy on each of such annuli. In the continuous case, the lower bound on each annulus is the straightforward estimate B R \B r w dx π deg(w, B R ) log R r, w H (B R \ B r ; S ). In Proposition 3. we prove a similar lower bound for F ε, with R/r replaced by R/(r+Cε log ε ), the error being due to the discrete structure of our energies. This weaker estimate, inserted in the ball construction machinery, is refined enough to prove the lower bound in Theorem 3. (ii). The second part of the paper is devoted to the analysis of metastable configurations for F ε and to our variational approach to the dynamics of discrete topological singularities. We now draw a parallel between the continuous Ginzburg-Landau model and our discrete systems, stressing out the peculiarities of our framework. In [33], [8], [39], it has been proved that the parabolic flow of GL ε can be described, as ε 0, by the gradient flow of the renormalized energy W(µ). Precisely the limiting flow is a measure µ(t) = M i= d i,0δ xi(t), where x(t) = (x (t),..., x M (t)) solves ẋ(t) = W (x(t)) (.3) π x(0) = x 0, with W (x(t)) = W(µ(t)). The advantage of this description is that the effective dynamics is described by an ODE involving only the positions of the singularities. This result has been derived through a purely variational approach in [39], based on the idea that the gradient flow structure is consistent with Γ-convergence, under some assumptions which imply that the slope of the approximating functionals converges to the slope of their Γ-limit. The gradient flow approach to dynamics used in the Ginzburg-Landau context fails for our discrete systems. In fact, the free energy of discrete systems is often characterized by the presence of many energy barriers, which affect the dynamics and are responsible for pinning effects (for a variational description of pinning effects in discrete systems see [3] and the references therein). As a consequence of our Γ-convergence analysis, we show that F ε has many local minimizers. Precisely, in Theorem 5.5 and Theorem 5.6 we show that, under suitable assumptions on the potential f, given any configuration of singularities x Ω M, there exists a stable configuration x at a distance of order ε from x. Starting from these configurations, the gradient flow of F ε is clearly stuck. Moreover, these stable configurations are somehow attractive wells for the dynamics. These results are proven for a general class of energies, including SD ε, while the case of the XY ε energy, to our knowledge, is still open. A similar analysis of stable configurations in the triangular lattice has been recently carried on in

5 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 5 [6], combining PDEs techniques with variational arguments, while our approach is purely variational and based on Γ-convergence. On one hand, our analysis is consistent with the well-known pinning effects due to energy barriers in discrete systems; on the other hand, it is also well understood that dislocations are able to overcome the energetic barriers to minimize their interaction energy (see [6], [], [7], [36]). The mechanism governing these phenomena is still matter of intense research. Certainly, thermal effects and statistical fluctuations play a fundamental role. Such analysis is beyond the purposes of this paper. Instead, we raise the question whether there is a simple variational mechanism allowing singularities to overcome the barriers, and then which would be the effective dynamics. We face these questions, following the minimizing movements approach à la De Giorgi ([5], [6], [9]). More precisely, we discretize time by introducing a time scale > 0, and at each time step we minimize a total energy, which is given by the sum of the free energy plus a dissipation. For any fixed, we refer to this process as discrete gradient flow. This terminology is due to the fact that, as tends to zero, the discrete gradient flow is nothing but the Euler implicit approximation of the continuous gradient flow of F ε. Therefore, as 0 it inherits the degeneracy of F ε, and pinning effects are dominant. The scenario changes completely if instead we keep fixed, and send ε 0. In this case, it turns out that, during the step by step energy minimization, the singularities are able to overcome the energy barriers, that are of order ε. Finally, sending 0 the solutions of the discrete gradient flows converge to a solution of (.3). In our opinion, this purely variational approach based on minimizing movements, mimics in a realistic way more complex mechanisms, providing an efficient and simple view point on the dynamics of discrete topological singularities in two dimensions. Summarizing, in order to observe an effective dynamics of the vortices we are naturally led to let ε 0 for a fixed time step, obtaining a discrete gradient flow of the renormalized energy. A technical issue is that the renormalized energy is not bounded from below, and therefore, in the step by step minimization we are led to consider local rather than global minimizers. Precisely, we minimize the energy in a δ neighborhood of the minimizer at the previous step. Without this care, already at the first step we would have the trivial solution µ = 0, corresponding to the fact that dipoles annihilate and the remaining singularities reach the boundary of the domain. Nevertheless, for small the minimizers do not touch the constraint, so that they are in fact true local minimizers. In order to discuss some mathematical aspects of the asymptotic analysis of discrete gradient flows as ε, 0, we need to clarify the specific choice of the dissipations we deal with. The canonical dissipation corresponding to continuous parabolic flow is clearly the L dissipation. On the other hand, once ε is sent to zero, we have a finite dimensional gradient flow of the renormalized energy, for which it is more natural to consider as dissipation the Euclidean distance between the singularities. This, for ε > 0, corresponds to the introduction of a -Wasserstein type dissipation, D, between the vorticity measures. For two Dirac deltas D is nothing but the square of the Euclidean distance of the masses (see Definition (6.4)). We are then led to consider also the discrete gradient flow with this dissipation. By its very definition D is continuous with respect to the flat norm and this makes the analysis as ε 0 rather simple and somehow instructive in order to face the more complex case of L dissipation.

6 6 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE We first discuss in details the discrete gradient flows with flat dissipation. To this purpose, it is convenient to introduce the functional F ε (µ), defined as the minimum of F ε (u) among all u whose vorticity measure µ(u) is equal to µ. We fix an initial condition µ 0 := M i= d i,0δ xi,0 with d i,0 = and a sequence µ ε,0 X ε satisfying µ ε,0 flat µ 0, F ε (µ ε,0 ) lim = π µ 0 (Ω). ε 0 log ε Then given δ > 0 and let ε, > 0 we define µ ε,k by the following minimization problem µ ε,k argmin F ε (µ) + πd (µ, µ ε,k ) } (.4) : µ X ε, µ µ ε,k flat δ with µ ε,0 = µ ε,0. As a direct consequence of our Γ-convergence analysis, in Theorem 6.7 we will show that, as ε 0, µ ε,k converges, up to a subsequence, to a solution µ k X to µ k argmin W(µ) + πd (µ, µ k ) M : µ = d i,0 δ xi, } µ µ flat k δ. After identifying the vorticity measure with the positions of its singularities, we get that the vortices x k of µ k satisfy the following finite-dimensional problem x k argmin W (x) + π x } M x k : x Ω M, x i x i,k δ. In Theorem 6.6 we show that this constrained scheme converges, as 0, to the gradient flow of the renormalized energy (.3), until a maximal time T δ. The proof of this fact follows the standard Euler implicit method, with some care to handle local rather than global minimization. Moreover, as δ 0, Tδ converges to the critical time T (see Definition 6.3), at which either a vortex touches the boundary or two vortices collapse. We now discuss the discrete gradient flow with the L dissipation. Once again, we consider a step by step minimization problem as in (.4), with µ µ ε,k flat replaced by v vε,k L / log. More precisely, u ε,k argmin F ε (u) + eπiu e πiu ε,k } L : µ(u) µ(u log ε,k ) flat δ. The prefactor / log in front of the dissipation can be viewed as a time reparametrization, on which we will comment later. The asymptotics of these discrete gradient flow as ε 0 relies again on a Γ- convergence analysis, which keeps memory also of the L limit v of the variable e πiuε. Under suitable assumptions on the initial data, in Theorem 7.3 we show that, as ε 0, the solutions u ε,k converge, up to a subsequence, to a solution to vk argmin W(v) + v v k L : v H log loc(ω \ M i=y i,k }; S ), } M Jv = d i,0 δ yi,k, Jv Jvk flat δ, i= i= i=

7 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 7 where Jv is the distributional Jacobian of v and W is the renormalized energy in terms of v (see Theorem 4.5); namely, min Jv=µ W(v) = W(µ). Now, we wish to send to zero. This step is much more delicate than in the case of flat dissipation. Indeed, it is at this stage that we adopt the abstract method [39], and exploit it in the context of minimizing movements instead of gradient flows. This method relies on the proof of two energetic inequalities; the first relates the slope of the approximating functionals with the slope of the renormalized energy; the second one relates the scaled L norm underlying the parabolic flow of GL ε with the Euclidean norm of the time derivative of the limit singularities. In our discrete in time framework, we adapt the arguments in [39] by replacing derivatives by finite differences. A heuristic argument to justify the prefactor / log is that it is the correct scaling for the canonical vortex x/ x. Indeed, given V R representing the vortex velocity, a direct computation shows that (.5) lim 0 log x x x V x V = π V. As a matter of fact, in order to get a non trivial dynamics in the limit, we have to accelerate the time scale as tends to zero. This feature is well known also in the parabolic flow of Ginzburg-Landau functionals. In this respect, we observe that our time scaling is expressed directly as a function of the time step, while for the functionals GL ε it depends on the only scale parameter of the problem, which is the length scale ε. The explicit computation in (.5) has not an easy counterpart for general solutions vk, and (.5) has to be replaced by more sophisticated estimates (see (7.7) and (7.56)). This point is indeed quite technical, and makes use of a lot of analysis developed in [39], [40]. In conclusion, we believe that this paper provides a better understanding of equilibria of discrete systems characterized by energy concentration, and contributes to the debate in the mathematical community over the microscopic mechanisms governing the dynamics of discrete topological singularities, as vortices in XY spin systems and dislocations in crystals. For the latter, richer models could be considered, with more realistic energy densities and dissipations, taking into account the specific material properties and the kinematic constraints of the crystal lattice. Our variational approach, rather than giving a complete analysis of a specific model, aims to be simple and robust, with possible applications to a wide class of discrete systems.. The discrete model for topological singularities In this Section we introduce the discrete formalism used in the analysis of the problem we deal with. We will follow the approach of [8]; specifically, we will use the formalism and the notations in [] (see also [35]). Let Ω R be a bounded open set with Lipschitz boundary, representing the domain of definition of the relevant fields in the models we deal with. The discrete lattice. For every ε > 0, we define Ω ε Ω as follows Ω ε := (i + εq), i εz : i+εq Ω where Q = [0, ] is the unit square. Moreover we set Ω 0 ε := εz Ω ε, and Ω ε := (i, j) Ω 0 ε Ω 0 ε : i j = ε, i j } (where i j means that i l j l for

8 8 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE l, }). These objects represent the reference lattice and the class of nearest neighbors, respectively. The cells contained in Ω ε are labeled by the set of indices Ω ε = i Ω 0 ε : i + εq Ω ε }. Finally, we define the discrete boundary of Ω as (.) ε Ω := Ω ε εz. In the following, we will extend the use of these notations to any given open subset A of R... Discrete functions and discrete topological singularities. Here we introduce the classes of discrete functions on Ω 0 ε, and a notion of discrete topological singularities. To this purpose, we first set AF ε (Ω) := u : Ω 0 ε R }, which represents the class of admissible scalar functions on Ω 0 ε. Moreover, we introduce the class of admissible fields from Ω 0 ε to the set S of unit vectors in R (.) AX Y ε (Ω) := v : Ω 0 ε S }, Notice that, to any function u AF ε (Ω), we can associate a function v AX Y ε (Ω) setting v = v(u) := e πiu. With a little abuse of notation for every v : Ω 0 ε R we denote (.3) v L = ε v(j). j Ω 0 ε Now we can introduce a notion of discrete vorticity corresponding to both scalar and S valued functions. To this purpose, let P : R Z be defined as follows (.4) P (t) = argmin t s : s Z}, with the convention that, if the argmin is not unique, then we choose the minimal one. Let u AF ε (Ω) be fixed. For every i Ω ε we introduce the vorticity (.5) α u (i) := P (u(i + εe ) u(i)) + P (u(i + εe + εe ) u(i + εe )) P (u(i + εe + εe ) u(i + εe )) P (u(i + εe ) u(i)). One can easily see that the vorticity α u takes values in, 0, }. Finally, we define the vorticity measure µ(u) as follows (.6) µ(u) := α u (i)δ i+ ε (e+e). i Ω ε This definition of vorticity extends to S valued fields in the obvious way, by setting µ(v) = µ(u) where u is any function in AF ε (Ω) such that v(u) = v. Let M(Ω) be the space of Radon measures in Ω and set } N X := µ M(Ω) : µ = d i δ xi, N N, d i Z \ 0}, x i Ω, (.7) i= X ε := µ X : µ = α(i)δ i+ ε (e+e), α(i), 0, }. i Ω ε

9 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 9 We will denote by µ n flat µ the flat convergence of µ n to µ, i.e., in the dual W, of W, 0... The discrete energy. Here we introduce a class of energy functionals defined on AF ε (Ω). We will consider periodic potentials f : R R which satisfy the following assumptions: For any a R () f(a + z) = f(a) for any z Z, () f(a) eπia = cos πa, (3) f(a) = π (a z) + O( a z 3 ) for any z Z. For any u AF ε (Ω), we define (.8) F ε (u) := (i,j) Ω ε f(u(i) u(j)). As explained in the Introduction, the main motivation for our analysis comes from the study discrete screw dislocations in crystals and XY spin systems. We introduce the basic energies for these two models as in []. Regarding the screw dislocations, for any u : Ω 0 ε R, we define (.9) SD ε (u) := dist (u(i) u(j), Z). (i,j) Ω ε It is easy to see that this potential fits (up to the prefactor 4π ) with our general assumptions. As for the XY model, for any v : Ω 0 ε S, we define (.0) XY ε (v) := v(i) v(j). (i,j) Ω ε Also this potential fits our framework, once we rewrite it in terms of the phase u of v. Indeed, setting f(a) = cos(πa), we have (.) XY ε (v) = f(u(i) u(j)) with v = e πiu. (i,j) Ω ε We notice that assumption () on F ε reads as (.) F ε (u) XY ε (e πiu ). Let T ± } i be the family of the ε-simplices of R whose vertices are of the form i, i ± εe, i ± εe }, with i εz. For any v : Ω 0 ε S, we denote by ṽ : Ω ε S the piecewise affine interpolation of v, according with the triangulation T ± } i. It is easy to see that, up to boundary terms, XY ε (v) corresponds to the Dirichlet energy of ṽ in Ω ε ; more precisely (.3) ṽ dx + Cε XY ε (v) ṽ dx, Ω ε Ω ε where C depends only on Ω. Remark.. Let v : Ω 0 ε S. One can easily verify that if A is an open subset of Ω and if ṽ > c > 0 on A ε, then (.4) µ(v)(a ε ) = deg(ṽ, A ε ),

10 0 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE where the degree of a function w H ( A; R ) with w c > 0, is defined by (.5) deg(w, A) := ( w π w w ) ds, w with v v := v v v v, for v H (A; R ). In particular, whenever ṽ > 0 on i + εq we have µ(v)(i + εq) = 0. A 3. Localized lower bounds In this section we will prove a lower bound for the energies F ε localized on open subsets A Ω. We will use the standard notation F ε (, A) (and as well XY ε (, A)) to denote the functional F ε defined in (.8) with Ω replaced by A. To this purpose, thanks to assumption () on the energy density f, it will be enough to prove a lower bound for the XY ε energy. As a consequence of this lower bound, we obtain a sharp zero-order Γ-convergence result for the functionals F ε. 3.. The zero-order Γ-convergence. We recall that the space of finite sums of weighted Dirac masses has been denoted in (.7) by X. Theorem 3.. The following Γ-convergence result holds. (i) (Compactness) Let u ε } AF ε (Ω) be such that F ε (u ε ) C log ε for some positive C. Then, up to a subsequence, µ(u ε ) flat µ, for some µ X. (ii) (Localized Γ-liminf inequality) Let u ε } AF ε (Ω) be such that µ(u ε ) flat µ = M i= d iδ xi with d i Z \ 0} and x i Ω. Then, there exists a constant C R such that, for any i =,..., M and for every σ < dist(x i, Ω j i x j), we have (3.) lim inf ε 0 F ε(u ε, B σ (x i )) π d i log σ ε C. In particular (3.) lim inf ε 0 F ε(u ε ) π µ (Ω) log σ ε C. (iii) (Γ-limsup inequality) For every µ X, there exists a sequence u ε } AF ε (Ω) such that µ(u ε ) flat µ and π µ (Ω) lim sup ε 0 F ε (u ε ) log ε. The above theorem has been proved in [35] for F ε = SD ε and in [] for F ε = XY ε, with (ii) replaced by the standard global Γ-liminf inequality (3.3) π µ (Ω) lim inf ε 0 F ε (u ε ) log ε, which is clearly implied by (3.). By (.), the compactness property (i) follows directly from the zero-order Γ- convergence result for the XY ε energies, while the proof of (ii) requires a specific analysis. For the convenience of the reader we will give a self contained proof of both (i) and (ii) of Theorem 3.. We will omit the proof of the Γ-lim sup inequality (iii) which is standard and identical to the XY ε case. Before giving the proof of Theorem 3., we need to revisit a construction referred to as ball construction and introduced in the continuous framework in [37], [9].

11 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 3.. Lower bound on annuli. Let w H (B R \ B r ; S ) with deg(w, B R ) = d. By Jensen s inequality, the following lower bound holds w dx R (w w) ds dρ B R \B r r B (3.4) ρ R ρ πd dρ π d log R r. r The latter is a key estimate in the context of continuous Ginzburg-Landau. In the following we will prove an analogous lower bound for the energy XY ε (v, ) in an annulus in which the piecewise affine interpolation ṽ satisfies ṽ. In view of (.) such a lower bound will hold also for the energy F ε. Proposition 3.. Fix ε > 0 and let ε < r < R ε. For any function v : (B R \ B r ) εz S with ṽ in B R ε \ B r+ ε, it holds (3.5) XY ε (v, B R \ B r ) π µ(v)(b r ) log where α > 0 is a universal constant. Proof. By (.3), using Fubini s theorem, we have that (3.6) XY ε (v, B R \ B r ) R ε r+ ε R r + ε ( α µ(v)(b r ) + ), B ρ ṽ ds dρ. Fix r + ε < ρ < R ε and let T be a simplex of the triangulation of the ε-lattice. Set γ T (ρ) := B ρ T, let γ T (ρ) be the segment joining the two extreme points of γ T (ρ) and let γ(ρ) = T γ T (ρ); then (3.7) ṽ ds = ṽ ds = ṽ T H (γ T (ρ)) B ρ T γ T (ρ) T ṽ T H ( γ T (ρ)) = ṽ ds. T Set m(ρ) := min γ(ρ) ṽ ; using Jensen s inequality and the fact that H ( γ(ρ)) H ( B ρ ) we get ṽ ds ( m (ρ) ṽ γ(ρ) γ(ρ) ṽ ṽ ) (3.8) ṽ ds m (ρ) ( ṽ H ( γ(ρ)) ṽ ṽ ) ds ṽ (3.9) m (ρ) π d ρ where we have set d := deg(ṽ, B ρ ) = µ(v)(b r ), which does not depend on ρ since ṽ /. Now, let T (ρ) be the simplex in which the minimum m(ρ) is attained. Without loss of generality we assume that T (ρ) = Tī for some ī εz. Let P one of the γ(ρ) γ(ρ)

12 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE points of γ(ρ) for which ṽ(p ) = m(ρ). By elementary geometric arguments, one can show that (3.0) ṽ ds α m (ρ), B ρ ε for some universal positive constant α. In view of (3.7), (3.9) and (3.0), for any r + ε < ρ < R ε we have ṽ ds m (ρ) π d α m (ρ) π d α B ρ ρ ε επ d + αρ. By this last estimate and (3.6) we get (3.) XY ε (v, B R \ B r ) π µ(v)(b r ) log ε( π α µ(v)(b r) ) + R ε( π α µ(v)(b r) + ) + r. Assuming, without loss of generality, α <, we immediately get (3.5) for α = π α Ball Construction. Here we introduce a construction referred to as ball construction, introduced in [37], [9]. Let B = B R (x ),..., B RN (x N )} be a finite family of pairwise disjoint balls in R and let µ = N i= d iδ xi with d i Z \ 0}. Let F be a positive superadditive set function on the open subsets of R, i.e., such that F (A B) F (A) + F (B), whenever A and B are open and disjoint. We will assume that there exists c > 0 such that (3.) F (A r,r (x)) π µ(b r (x)) log R c + r, for any annulus A r,r (x) = B R (x) \ B r (x), with A r,r (x) Ω \ i B R i (x i ). The purpose of this construction is to select a family of larger and larger annuli in which the main part of the energy F concentrates. Let t be a parameter which represents an artificial time. For any t > 0 we want to construct a finite family of balls B(t) which satisfies the following properties () N i= B R i (x i ) B B(t) B, () the balls in B(t) are pairwise disjoint, (3) F (B) π µ(b) log( + t) for any B B(t), (4) B B(t) R(B) (+t) i R i +(+t)cn(n +N +), where R(B) denotes the radius of the ball B. We construct the family B(t), closely following the strategy of the ball construction due to Sandier and Jerrard, that we need to slightly revise in order to include our case: The only difference in our discrete setting is the appearance of the error term c > 0 in (3.) and in (4), while in the continuous setting c = 0. The ball construction consists in letting the balls alternatively expand and merge each other as follows. It starts with an expansion phase if dist(b Ri (x i ), B Rj (x j )) > c for all i j, and with a merging phase otherwise. Assume that the first phase is an expansion. It consists in letting the balls expand, without changing theirs centers, in such a way that, at each (artificial) time, the ratio θ(t) := Ri(t) c+r i is independent of i. We will parametrize the time enforcing θ(t) = + t. Note that with this choice R i (0) = R i + c so that the balls B Ri(0)(x i )} are pairwise disjoint. The first expansion phase stops at the first time T when two balls bump into each other. Then the merging phase begins. It consists in identifying a suitable partition Sj } j=,...,n n of the family B Ri(T )(x i ) }, and, for each subclass Sj, in finding a

13 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 3 ball B R j (x j ) which contains all the balls in S j hold: such that the following properties i) for every j k, dist(b R j (x j ), B Rk (x k )) > c; ii) Rj Nc is not larger than the sum of all the radii of the balls B R i(t )(x i ) Sj, i.e., contained in B Rj (x j ). After the merging, another expansion phase begins, during which we let the balls B R j (x j ) } expand in such a way that, for t T, for every j we have (3.3) R j (t) c + R j = + t + T. Again note that Rj (T ) = Rj +c. We iterate this process obtaining a set of merging times T,..., T n }, and a family B(t) = B R k j (t)(x k j )} j for t [T k, T k+ ), for all k =,..., n. Notice that n N. If the condition dist(b Ri (x i ), B Rj (x j )) > c for all i j, is not satisfied we clearly can start this process with a merging phase (in this case T = 0). By construction, we clearly have () and (). We now prove (4). Set N(t) = B B(t)} and I(t) =,..., N(t)}. Moreover, for every merging time T k and j N(T k ), set I j (T k ) := i I(T k ) : B R k i By ii) and (3.3) it follows that for any k n (3.4) N(T k ) (Rj k Nc) j= = = N(T k ) j= N(T k ) j= l I j(t k ) } (x k i ) B R k j (x k j ). R k l (T k ) ( + Tk + T k R k j + + T k + T k c N(T k ) + T k R k j + + T k cn(t k ) + T k + T k j= N(T k ) + T k + T k j= R k j + ( + T k )cn. Let T k t < T k+ for some k n; by (3.3) and iterating (3.4) we get (3.5) N(T k ) j= Rj k (t) = + t + T k ( + t) N(T k ) j= R k j + + t + T k c N(T k ) ) N R i + ( + t)cn(n + N + ), i= and this concludes the proof of (4). It remains to prove (3). For t = 0 it is trivially satisfied. We will show that it is preserved during the merging and the expansion times. Let T k be a merging time

14 4 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE and assume that (3) holds for all t < T k. Then for every j I(T k ) F (B R k j (x k j )) F (B R k l (T k ) (xk l )) l I j(t k ) π log( + T k ) j l= π log( + T k ) µ(b R k j (x k j )). µ(b R k l (T k ) (xk l )) Finally, for a given t [T k, T k+ ) and for any ball B R k i (t)(x k i (t)) B(t) we have F (B R k i (t)(x k i )) F (B R k i (t)(x k i ) \ B R k i (x k i )) + F (B R k i (x k i )) π µ(b R k i (t)(x k i )) log + t + T k + π µ(b R k i (t)(x k i )) log( + T k ) where we have used that Rk i (t) c+r k i = +t +T k. = π µ(b R k i (t)(x k i )) log( + t), 3.4. Proof of Theorem 3.. First, we give an elementary lower bound of the energy localized on a single square of the lattice, whose proof is immediate. Proposition 3.3. There exists a positive constant β such that for any ε > 0, for any function v AX Y ε (Ω) and for any i Ω ε such that the piecewise affine interpolation ṽ of v satisfies min i+εq ṽ <, it holds XY ε(v, i + εq) β. Proof of Theorem 3.. By (.), it is enough to prove (i) and (ii) for F ε = XY ε, using as a variable v ε = e πiuε. The proof of (iii) is standard and left to the reader. Proof of (i). For every ε > 0, set I ε := i Ω ε : min i+εq ṽ ε }. Notice that, in view of Remark., µ(v ε ) is supported in I ε + ε (e + e ). Starting from the family of balls B ε (i + ε (e + e ))), and eventually passing through a merging procedure we can construct a family of pairwise disjoint balls B ε := B Ri,ε (x i,ε ) } i=,...,n ε, with N ε i= R i,ε ε I ε. Then, by Proposition 3.3 and by the energy bound, we immediately have that I ε C log ε and hence (3.6) N ε i= We define the sequence of measures R i,ε εc log ε. N ε µ ε := µ(v ε )(B Ri,ε (x i,ε ))δ xi,ε. i= Since µ ε (B) I ε for each ball B B ε, by (3.5) we deduce that (3.) holds with F ( ) = XY ε (v ε, \ B Bε B) and c = ε(α I ε + ). We let the balls in the families B ε grow and merge as described in Subsection 3.3, and let B ε (t) := B Ri,ε(t)(x i,ε (t)) } be the corresponding family of balls at time t.

15 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 5 Set moreover t ε := ε, N ε (t ε ) := B ε (t ε ) and define (3.7) ν ε := µ ε (B Ri,ε(t ε)(x i,ε (t ε )))δ xi,ε(t ε). i=,...,n ε(t ε) B Ri,ε (tε)(x i,ε(t ε)) Ω By (3) in Subsection 3.3, for any B B ε (t ε ), with B Ω, we have XY ε (v ε, B) π µ ε (B) log( + t ε ) = π ν ε (B) log ε ; by the energy bound, we have immediately that ν ε (Ω) M and hence ν ε } is precompact in the weak topology. By (4) in Subsection 3.3, it follows that N ε(t ε) j= R j (t ε ) C ε ( I ε ) 4, which easily implies that ν ε µ ε flat 0; moreover, using (3.6), it is easy to show that µ ε µ(v ε ) flat 0 as ε 0 (see [3] for more details). We conclude that also µ(v ε ) is precompact in the flat topology. Proof of (ii). Fix i,..., M}. Without loss of generality, and possibly extracting a subsequence, we can assume that (3.8) lim inf ε 0 XY ε(v ε, B σ (x i )) π d i log ε = lim ε 0 XY ε (v ε, B σ (x i )) π d i log ε < +. We consider the restriction v ε AX Y ε (B σ (x i )) of v ε to B σ (x i ). Notice that supp(µ( v ε ) µ(v ε ) B σ (x i )) B σ (x i ) \ B σ ε (x i ). On the other hand, by (3.8) and Proposition 3.3 it follows that (3.9) µ(v ε ) (B σ (x i ) \ B σ ε (x i )) C log ε. Then, using (3.9) one can easily get (3.0) µ( v ε ) µ(v ε ) B σ (x i ) flat 0, and hence (3.) µ( v ε ) d i δ xi flat 0. We repeat the ball construction procedure used in the proof of (i) with Ω replaced by B σ (x i ), v ε by v ε and I ε by I i,ε := j (B σ (x i )) ε : min ṽ ε }. j+εq We denote by B i,ε the corresponding family of balls and by B i,ε (t) the family of balls constructed at time t. Fix 0 < γ < such that (3.) ( γ)( d i + ) > d i. Let t ε,γ = ε γ and let ν ε,γ be the measure defined as in (3.7) with Ω replaced by B σ (x i ) and t ε replaced by t ε,γ. As in the previous step, since γ > 0 we deduce that ν ε,γ d i δ xi flat 0; moreover, for any B B i,ε (t ε,γ ) we have (3.3) XY ε (v ε, B) π ν ε,γ (B) ( γ) log ε. Now, if lim inf ε 0 ν ε,γ (B σ (x i )) > d i, then, thanks to (3.), (3.) holds true. Otherwise we can assume that ν ε,γ (B σ (x i )) = d i for ε small enough. Then

16 6 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE ν ε,γ is a sum of Dirac masses concentrated on points which converge to x i, with weights all having the same sign and summing to d i. Let C > 0 be given and set σ t ε := C ( I i,ε) 4 ε. By (3.5), we have that any ball B B i,ε( t ε ) satisfies diam(b) C C σ, where C > 0 is a universal constant. We fix C > C so that diam(b) < σ. Recall that, for ε small enough, supp(ν ε,γ ) B σ/ (x i ); hence if B B i,ε ( t ε ) with supp(ν ε,γ ) B, then B B σ (x i ) and one can easily show that ( ) µ( v ε ) B = d i. We have immediately that XY ε ( v ε, B σ (x i ) \ B Bi,ε B) π B B i,ε( t ε) B B σ(x i) B B i,ε( t ε) B B σ(x i) µ( v ε )(B) log( + t ε ) π d i log σ C ( I i,ε ) 4 ε. On the other hand, by Proposition 3.3 there exists a positive constant β such that XY ε ( v ε, j + εq) β for any j I i,ε ; therefore, XY ε ( v ε, B B i,ε B) β I i,ε. Finally, we get XY ε ( v ε, B σ (x i )) XY ε ( v ε, B σ (x i ) \ B Bi,ε B) + XY ε ( v ε, B Bi,ε B) π d i log σ ε log ( C ( I i,ε ) 4) + I i,ε β π d i log σ ε + C and (3.) follows sending ε The renormalized energy and the first order Γ-convergence. In this section we will prove the first order Γ-convergence of F ε to the renormalized energy, introduced in the continuous framework of Ginzburg-Landau energies in [0]. To this purpose we begin by recalling the many definitions and results of [0] we need. 4.. Revisiting the analysis of Bethuel-Brezis-Hélein. Fix µ = M i= d iδ xi with d i, +} and x i Ω. In order to define the renormalized energy, consider the following problem Φ = πµ in Ω Φ = 0 on Ω, and let R 0 (x) = Φ(x) M i= d i log x x i. Notice that R 0 is harmonic in Ω and R 0 (x) = M i= d i log x x i for any x Ω. The renormalized energy corresponding to the configuration µ is then defined by (4.) W(µ) := π i j d i d j log x i x j π i d i R 0 (x i ). Let σ > 0 be such that the balls B σ (x i ) are pairwise disjoint and contained in Ω and set Ω σ := Ω \ M i= B σ(x i ). A straightforward computation shows that (4.) W(µ) = lim Φ dx Mπ log σ, σ 0 Ω σ

17 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 7 In this respect the renormalized energy represents the finite energy induced by µ once the leading logarithmic term has been removed. It is convenient to consider (as done in [0]) suitable cell problems and auxiliary minimum problems. Set (4.3) m(σ, µ) := min w H (Ω σ ;S ) m(σ, µ) := min w H (Ω σ ;S ) } w dx : deg(w, B σ (x i )) = d i, Ω σ Ω σ w dx : w( ) = α i σ di ( x i) di on B σ (x i ), α i = For any x R \ 0}, we define θ(x) as the polar coordinate arctan x /x, also referred to as the lifting of the function x x. Given ε > 0 we introduce a discrete minimization problem in the ball B σ (4.4) γ(ε, σ) := min u AF ε(b σ) F ε(u, B σ ) : πu( ) = θ( ) on ε B σ }, where the discrete boundary ε is defined in (.). Theorem 4.. It holds (4.5) lim σ 0 m(σ, µ) π µ (Ω) log σ = lim σ 0 m(σ, µ) π µ (Ω) log σ = W(µ). Moreover, for any fixed σ > 0, the following limit exits finite (4.6) lim(γ(ε, σ) π log ε ) =: γ R. ε 0 σ The proof of (4.5) is contained in [0], whereas the statement in (4.6) is a discrete version of Lemma III. in [0] and can be proved similarly. We give the details of the proof of (4.6) for the convenience of the reader. Proof of (4.6). First, by scaling, it is easy to see that γ(ε, σ) = I( ε σ ) where I(t) is defined by } I(t) := min F (θ, B ) πu = θ on B. t t We aim to prove that (4.7) 0 < t t I(t ) π log t t + I(t ) + O(t ). Notice that by (4.7) it easily follows that lim t 0 +(I(t) π log t ) exists and is not +. Moreover, by Theorem 3., there exists a universal constant C such that I(t) π log t + C t (0, ]. We conclude that lim t 0 +(I(t) π log t ) is not. In order to complete the proof we have to show that (4.7) holds. To this end, set A r,r = B R \ B r, and let θ be the lifting of the function x x. Since θ(x) c/r for every x A r,r, by standard interpolation estimates (see for instance [9]) and using assumption (3) on f, we have that, as r < R, (4.8) F (θ/π, A r,r ) π log R r + O(/r). }.

18 8 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE Let u be a minimizer for I(t ) and for any i Z define By (4.8) we have I(/R) (i,j) (B r) i,j (B r) u (i) := u (i) if i t θ(i) π if t i t, f(u (i) u (j)) + (i,j) (A r,r ) i,j (A r,r ) f(u (i) u (j)) I(/r) + π log r R + O(/r), which yields (4.7) for r = t and R = t. 4.. The main Γ-convergence result. We are now in a position to state the first-order Γ-convergence theorem for the functionals F ε. Theorem 4.. The following Γ-convergence result holds. (i) (Compactness) Let M N and let u ε } AF ε (Ω) be a sequence satisfying F ε (u ε ) Mπ log ε C. Then, up to a subsequence, µ(u ε ) flat µ for some µ = N i= d iδ xi with d i Z \ 0}, x i Ω and i d i M. Moreover, if i d i = M, then i d i = N = M, namely d i = for any i. (ii) (Γ-lim inf inequality) Let u ε } AF ε (Ω) be such that µ(u ε ) flat µ, with µ = M i= d iδ xi with d i = and x i Ω for every i. Then, (4.9) lim inf F ε(u ε ) Mπ log ε W(µ) + Mγ. ε 0 (iii) (Γ-lim sup inequality) Given µ = M i= d iδ xi with d i = and x i Ω for every i, there exists u ε } AF ε (Ω) with µ(u ε ) flat µ such that F ε (u ε ) Mπ log ε W(µ) + Mγ. In our analysis it will be convenient to introduce the energy functionals F ε in term of the variable µ, i.e., by minimizing F ε with respect to all u AF ε (Ω) with µ(u) = µ. Precisely, let F ε : X [0, + ] be defined by (4.0) F ε (µ) := inf F ε (u) : u AF ε (Ω), µ(u) = µ}. Theorem 4. can be rewritten in terms of F ε as follows. Theorem 4.3. The following Γ-convergence result holds. (i) (Compactness) Let M N and let µ ε } X be a sequence satisfying flat F ε (µ ε ) Mπ log ε C. Then, up to a subsequence, µ ε µ = N i= d iδ xi with d i Z \ 0}, x i Ω and i d i M. Moreover, if i d i = M, then i d i = N = M, namely d i = for every i. flat (ii) (Γ-lim inf inequality) Let µ ε } X be such that µ ε µ = M i= d iδ xi with d i = and x i Ω for every i. Then, (4.) lim inf F ε(µ ε ) Mπ log ε W(µ) + Mγ. ε 0

19 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES 9 (iii) (Γ-lim sup inequality) Given µ = M i= d iδ xi with d i = and x i Ω for flat every i, there exists µ ε } X with µ ε µ such that (4.) F ε (µ ε ) Mπ log ε W(µ) + Mγ The proof of Theorem 4.. Recalling that F ε (u) XY ε (e πiu ), the proof of the compactness property (i) will be done for F ε = XY ε, and will be deduced by Theorem 3.. On the other hand, the constant γ in the definition of the Γ- limit depends on the details of the discrete energy F ε, and its derivation requires a specific proof. Proof of (i): Compactness. The fact that, up to a subsequence, µ(u ε ) flat µ = N i= d iδ xi with N i= d i M is a direct consequence of the zero order Γ-convergence result stated in Theorem 3. (i). Assume now N i= d i = M and let us prove that d i =. Let 0 < σ < σ be such that B σ (x i ) are pairwise disjoint and contained in Ω and let ε be small enough so that B σ (x i ) are contained in Ω ε. For any 0 < r < R and x R, set A r,r (x) := B R (x) \ B r (x). Since F ε (u ε ) XY ε (e πiuε ), N N (4.3) F ε (u ε ) XY ε (e πiuε, B σ (x i )) + XY ε (e πiuε, A σ,σ (x i )). i= To ease notation we set v ε = e πiuε and we indicate with ṽ ε the piecewise affine interpolation of v ε. Moreover let t be a positive number and let ε be small enough so that t > ε. Then, by (3.) and (.3), we get N F ε (u ε ) π d i log σ ε + N (4.4) ṽ ε dx + C. i= i= i= A σ +t,σ t(x i) By the energy bound, we deduce that A σ +t,σ t(x i) ṽ ε dx C and hence, up to a subsequence, ṽ ε v i in H (A σ+t,σ t(x i ); R ) for some field v i. Moreover, since ε ( ṽ ε ) dx CXY ε (v ε ) C log A σ +t,σ t(x i) ε, (see Lemma in [] for more details ), we deduce that v i = a.e. Furthermore, by standard Fubini s arguments, for a.e. σ + t < σ < σ t, up to a subsequence the trace of ṽ ε is bounded in H ( B σ (x i ); R ), and hence it converges uniformly to the trace of v i. By the very definition of degree it follows that deg(v i, B σ (x i )) = d i. Hence, by (3.4), for every i we have (4.5) A σ +t,σ t(x i) v i dx d i π log σ t σ + t. By (4.4) and (4.5), we conclude that for ε sufficiently small N ( F ε (u ε ) π d i log σ ε + d i log σ ) t + C σ + t Mπ log ε + π i= i= N ( d i d i ) log σ N + π d i log σ (σ t) σ σ (σ + t) + C. i=

20 0 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE The energy bound yields that the sum of the last two terms is bounded; letting t 0 and σ 0, we conclude d i =. Proof of (ii): Γ-liminf inequality. Fix r > 0 so that the balls B r (x i ) are pairwise disjoint and compactly contained in Ω. Let moreover Ω h} be an increasing sequence of open smooth sets compactly contained in Ω such that h N Ω h = Ω. Without loss of generality we can assume that F ε (u ε ) Mπ log ε + C, which together with Theorem 3. yields M (4.6) F ε (u ε, Ω \ B r (x i )) C. i= We set v ε := e πiuε and we denote by ṽ ε the piecewise affine interpolation of v ε. For every r > 0, by (4.6) and by (.) we deduce XY ε (v ε \ N i= B r(x i )) C. Fix h N and let ε be small enough so that Ω h Ω ε. Then, Ω h \ ṽ ε dx C; N i= Br(xi) therefore, by a diagonalization argument, there exists a unitary field v with v H (Ω \ M i= B ρ(x i ); S ) for any ρ > 0 and a subsequence ṽ ε } such that ṽ ε v in Hloc (Ω \ M i= x i}; R ). Let σ > 0 be such that B σ (x i ) are pairwise disjoint and contained in Ω h. For any 0 < r < R < + and for any x R, set A r,r (x) := B R (x) \ B r (x), A r,r := A r,r (0). Let t σ, and consider the minimization problem min w H (A t/,t ;S ) A t/,t w dx : deg(w, B t ) = It is easy to see that the minimum is π log and that the set of minimizers is given by (the restriction at A t/,t of the functions in) (4.7) K := α z } : α C, α =. z Set (4.8) d t (w, K) := min w v L (At/,t;R ) : v K }. It is easy to see that for any given δ > 0 there exists a positive ω(δ) (independent of t) such that if d t (ṽ ε ( + x i ), K) δ, then (4.9) lim inf ṽ ε dx π log + ω(δ). ε 0 A t + ε,t ε (xi) By a scaling argument we can assume t =. Then, arguing by contradiction, if there exists a subsequence ṽ ε } such that lim ṽ ε dx = π log, ε 0 A + ε, ε (xi) then, by the lower semicontinuity of the L norm, we get (4.0) π log v dx lim ṽ ε dx = π log. A /, (x ε 0 i) A + ε, ε (xi) }.

21 METASTABILITY AND DYNAMICS OF DISCRETE TOPOLOGICAL SINGULARITIES It follows that v( + x i ) K, and that ṽ ε v strongly in H (A /, (x i ); R ), which yields the contradiction dist(v( + x i ), K) δ. Let L N be such that L ω(δ) W(µ) + M(γ π log σ C) where C is the constant in (3.). For l =,..., L, set C l (x i ) := B l σ(x i ) \ B l σ(x i ). We distinguish among two cases. First case: for ε small enough and for every fixed l L, there exists at least one i such that d l σ(ṽ ε ( + x i ), K) δ. Then, by (3.), (4.9) and the lower semicontinuity of the L norm, we conclude F ε (u ε, Ω h ) M XY ε (v ε, B L σ(x i )) + i= L l= i= M XY ε (v ε, C l (x i )) M(π log σ + π log ε + C) + L(Mπ log + ω(δ)) + o(ε) L Mπ log ε + Mγ + W(µ) + o(ε). Second case: Up to a subsequence, there exists l L such that for every i we have d σ (ṽ ε ( + x i ), K) δ, where σ := lσ. Let α ε,i be the unitary vector x x such that ṽ ε α i ε,i x x i H (C l(x i);r ) = d σ (ṽ ε ( + x i ), K). One can construct a function ū ε AF ε (Ω) such that (i) ū ε = u ε on ε (R \ B lσ (x i)); (ii) e πiūε = α ε e iθ on ε B lσ (x i) (iii) F ε (u ε, B σ (x i )) F ε (ū ε, B σ (x i )) + r(ε, δ) with lim δ 0 lim ε 0 r(ε, δ) = 0. The proof of (i)-(iii) is quite technical, and consists in adapting standard cut-off arguments to our discrete setting. For the reader convenience we skip the details of the proof, and assuming (i)-(iii) we conclude the proof of the lower bound. By Theorem (4.), we have that M M F ε (u ε ) XY ε (v ε, Ω h \ B σ (x i )) + F ε (u ε, B σ (x i )) M Ω h \ ṽ ε dx + F ε (ū ε, B σ (x i )) + r(ε, δ) + o(ε) M i= B σ(xi) i= Ω h \ ṽ ε dx + M(γ π log ) + r(ε, δ) + o(ε) M ε σ i= B σ(xi) Ω\ v dx + M(γ π log ) + r(ε, δ) + o(ε) + o(/h) M ε σ i= B σ(xi) Mπ log ε + Mγ + W(µ) + r(ε, δ) + o(ε) + o( σ) + o(/h). The proof follows sending ε 0, δ 0, σ 0 and h. Proof of (iii): Γ-limsup inequality. This proof is standard in the continuous case, and we only sketch its discrete counterpart. Let w σ be a function that agrees with a minimizer of (4.3) in Ω \ M i= B σ(x i ) =: Ω σ x x. Then, w σ = α i i σ on B σ (x i ) for some α i =. For every ρ > 0 we can always find a function w σ,ρ C (Ω σ ; S ) such that x x w σ,ρ = α i i σ on B σ (x i ), and w σ,ρ dx w σ dx ρ. Ω σ Ω σ i= i=

22 R. ALICANDRO, L. DE LUCA, A. GARRONI, AND M. PONSIGLIONE Moreover, for every i let w i AX Y ε (B σ (x i )) be a function which agrees with x x α i i x x on i εb σ (x i ) and such that its phase minimizes problem (4.4). If necessary, we extend w i to (B σ (x i ) εz ) \ (B σ (x i )) 0 x x ε to be equal to α i i x x i. Finally, define the function w ε,σ,ρ AX Y ε (Ω) which coincides w σ,ρ on Ω σ εz and with w i on B σ (x i ) εz. Then, in view of assumption (3) on f, a straightforward computation shows that any phase u ε,σ,ρ of w ε,σ,ρ is a recovery sequence, i.e., lim F ε(u ε,σ,ρ ) Mπ log ε = Mγ + W(µ) + o(ρ, σ), ε 0 with lim σ 0 lim ρ 0 o(ρ, σ) = Γ-convergence analysis in the L topology. Here we prove a Γ-convergence result for F ε (u ε ) Mπ log ε, where M is fixed positive integer, with respect to the flat convergence of µ(u ε ) and the L -convergence of ṽ ε, where ṽ ε : Ω ε R is the piecewise affine interpolation of e πiuε. To this purpose, for N N let us first introduce the set (4.) D N := v L (Ω; S ) : Jv = π N d i δ xi with d i =, x i Ω, i= v H loc(ω \ supp(jv); S )}. Notice that, if v D M, then the function Ω\ v dx Mπ log σ, M i= Bσ(xi) is monotonically decreasing with respect to σ. functional W : L (Ω; S ) R given by (4.) W(v) = lim σ 0 Therefore, it is well defined the Ω\ M i= Bσ(xi) v dx Mπ log σ if v D M ; if v D N for some N < M; + otherwise Notice that, by (4.5) we have that, for every µ = M i= d iδ xi with d i = (4.3) W(µ) = min W(v). v H loc (Ω\supp(µ);S ) Jv=µ Remark 4.4. We can rewrite W(v) as follows W(v) = Ω\ ib ρ(x i) v dx + Mπ log ρ + M + i= j=0 ( ) v dx π log, C i,j where C i,j denotes the annulus B j ρ(x i ) \ B ρ(x (j+) i ). In particular, for the lower bound (3.4) we deduce that (4.4) sup v dx π log + W(v) Mπ log ρ. i,j C i,j