Topological Singularities in Periodic Media: Ginzburg–Landau and Core-Radius Approaches

We describe the emergence of topological singularities in periodic media within the Ginzburg–Landau model and the core-radius approach. The energy functionals of both models are denoted by Eε,δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\varepsilon ,\delta }$$\end{document}, where ε\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon $$\end{document} represent the coherence length (in the Ginzburg–Landau model) or the core-radius size (in the core-radius approach) and δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta $$\end{document} denotes the periodicity scale. We carry out the Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-convergence analysis of Eε,δ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$E_{\varepsilon ,\delta }$$\end{document} as ε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon \rightarrow 0$$\end{document} and δ=δε→0\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\delta =\delta _\varepsilon \rightarrow 0$$\end{document} in the |logε|\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$|\log \varepsilon |$$\end{document} scaling regime, showing that the Γ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\Gamma $$\end{document}-limit consists in the energy cost of finitely many vortex-like point singularities of integer degree. After introducing the scale parameter λ=min{1,limε→0|logδε||logε|}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} \lambda =\min \Bigl \{1,\lim _{\varepsilon \rightarrow 0} {|\log \delta _\varepsilon |\over |\log \varepsilon |}\Bigr \} \end{aligned}$$\end{document}(upon extraction of subsequences), we show that in a sense we always have a separation-of-scale effect: at scales smaller than ελ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^\lambda $$\end{document} we first have a concentration process around some vortices whose location is subsequently optimized, while for scales larger than ελ\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\varepsilon ^\lambda $$\end{document} the concentration process takes place “after” homogenization.


Introduction
Phase transitions mediated by the formation of topological defects characterize several physical phenomena such as superfluidity, superconductivity and plasticity (see [39,40,41,31,37,32]). The study of such topological defects has become an extremely active research field in mathematics after the progresses achieved in the analysis of the Ginzburg Landau (GL) energy functional in the last decades (see e.g.[13,45]).In [3] it has been proved that the GL functional, originally introduced to model the phenomenology of phase transitions in Type-II superconductors through the formation of vortex singularities of a complex order parameter, provides a good variational description for the emergence of vortices in XY spin systems and of screw dislocations in crystal plasticity (see also [2,42,4,24,9]).The results obtained in [3] suggest to exploit the GL theory for a phenomenological 1 alternative description of several material-dependent variational models, opening the way to a number of new mathematical problems involving the analysis of this functional.For instance, in the modeling of materials, one needs to suitably modify it to include the usual kinematic constraints and material constants which are specific of crystal structures.As a first step in this direction, here we study a variant of the GL energy functional to include heterogeneities of the medium.
Before describing the case of heterogeneous media, we briefly recall the analysis in the homogeneous case.Let Ω ⊂ R 2 be an open bounded set and let ε denote the coherence length of the GL energy (proportional to the length scale of the core of a screw dislocation in a plastic crystal or to the lattice spacing in a XY spin system).Let a > 0 and let GL ε : H 1 (Ω; R 2 ) → R be the Ginzburg-Laundau functional defined as (0.1) The asymptotic behavior of GL ε as ε → 0 has been studied in order to give an energetic description of the onset of vortices (see for instance [13,45]).A prototypical vortex of degree z ∈ Z \ {0} at a point x 0 ∈ Ω can be thought of as the point singularity of a vectorial order parameter vε : Ω → R 2 which, outside the ball of radius ε centered at x 0 , winds around the center as ( x−x0 |x−x0| ) z .The energy of vε diverges at order | log ε| as ε → 0. As a consequence, to detect the effective energy cost of finitely many vortex singularities, one needs to study the GL ε energy at a logarithmic scaling; that is, to consider the asymptotic behavior of functionals GL ε (v)  | log ε| .It has been proved in [34,1] that a sequence {v ε }, along which these energy functionals are equi-bounded, has Jacobians Jv ε that, up to a subsequence, converge in the flat sense (see Section 1) to an atomic measure µ = n i=1 z i δ xi whose support represents the position of the limiting vortices.The Γ-limit of GL ε | log ε| with respect to this convergence at µ is then given by 2πa n i=1 |z i | (supposing x i = x j if i = j).This value can be rewritten as 2πa|µ|(Ω) and thought of as a functional depending on the total variation |µ|(Ω) of µ in Ω.Now, if more in general Ω is regarded as a reference configuration of a heterogeneous material, described by periodic heterogeneities at a length scale δ ε , we may consider the energies GL ε : where a : R 2 → [α, β] (0 < α < β) is a (0, 1) 2 -periodic function describing the material properties of the media.Note that the energy GL ε is controlled from (above and) below by a multiple of the GL energy GL ε above.Therefore, setting the following compactness result holds true.
Assuming δ ε → 0 as ε → 0 we expect the effective limiting energy at the vortex scaling to be a homogeneous energy combining both homogenization and concentration effects.As these effects depend on the mutual rate of convergence of the vanishing parameters ε and δ ε , different regimes are possible.Heuristically, at some extreme regimes we will have "separation of scales".Namely, if ε tends to 0 sufficiently fast with respect to δ = δ ε then we expect that δ can be thought of as an independent variable, the dependence on which separately dealt with after letting ε → 0 with fixed δ.In this case, the limit as ε → 0 with δ fixed gives an energy of the form 2π n i=1 |z i |a xi δ , and the optimization of the location of vortices at minimum points for a (we may assume here that a be continuous), which tend to be dense as δ → 0, finally provides a limit of the form 2π min a n i=1 |z i |.
Note that in order that this argument may work, the energy of a recovery sequence should be concentrated on a o(δ)-neighborhood of a minimum point of a.This gives a condition by testing with functions winding as x−xi |x−xi| around a vortex x i .Conversely, if δ = δ ε tends to 0 sufficiently fast with respect to ε, we expect that the variable ε be considered as fixed and a homogenization process may be first performed with δ → 0. In this case, moreover, since the potential term in (0.2) forces v ε to have modulus equal to one as ε → 0, (neglecting for a moment the effect of singularities) we may regard the homogenization process to be restricted to the first part of the energy in (0.2), which can be written as where u is the lifting of v, i.e., v = e ıu .The homogenization of functionals of this form has been extensively studied in terms of Γ-convergence (see [19]) and it has been shown that G δ Γ −→ G 0 as δ → 0 , where G 0 (u) := Ω A hom ∇u, ∇u dx, and A hom is the two-by-two symmetric matrix defined by (0.4) A hom ξ, ξ := inf (0,1) 2 a(y)|ξ + ∇ϕ(y)| 2 dy : ϕ ∈ W 1,∞ per ((0, 1) 2 ) .
At this point, the subsequent analysis involves the study of the Γ-limit as ε → 0 of a homogeneous but anisotropic energy functional related to G 0 at scale | log ε|.The validity of this separation of scales can be formalized by using a coarea formula-type argument, which shows that the Γconvergence of GL ε can be obtained working within another well-known framework in the analysis of topological singularities; i.e., the so-called core-radius approach.That approach consists in computing the gradient term in the energy outside small regions -the cores -around the singularities, and allows to directly work with S 1 -valued order parameters (see e.g.[13,5]).In this framework, we may describe the energy around a vortex of degree z by an asymptotic formula of the type (0.5) ψ(z) = lim In order that this argument work, minimum problems in (0.5) should be seen as limits of minimum problems It is interesting to note that there is a scale gap between the two separation-of-scale regimes given by (0.3) and (0.7); i.e., when (the existence of the limit is not restrictive up to extraction of subsequences).In this case the behavior of the Γ-limit is a convex combination of the extreme ones.Recovery sequences are constructed with vortices concentrating close to minimum points for a, while they optimize oscillations at scales between ε and 1 so as to obtain a homogenized overall behavior at those scales.
The final form of the Γ-limit is then which comprises also the extreme cases, upon setting (0.9) Note that, since in the logarithmic regime, the GL ε energies concentrate at any scale between ε and 1, their behavior is very different from that of the corresponding scalar version, the inhomogeneous Cahn-Hilliard functionals given (after scaling) by which concentrate at scale ε producing sharp-interface models.In that case separation of scale occurs for ε ≪ δ ε and δ ε ≪ ε, while in the critical regime δ ε ∼ ε the effective surface tension is described by an optimal-profile problem depending on K := lim ε→0 δ ε /ε (see [6]).In a sense, in the GL case we do not have a critical behavior and we always have separation of scales.The parameter λ above can be seen as describing a threshold scale above and below which the two types of separation of scales take place.
Although suggested by the heuristics, the computation of the Γ-limits described above is highly non-trivial and needs several new ideas in order to combine techniques from GL and homogenization theories.We briefly outline some of the most relevant technical issues, and state the main results of the paper formalizing the heuristic description given above, subdividing the analysis in the cases δ ε ε and δ ε ≫ ε The following result is proven in Section 6.2.
(ii) ( Γ-limsup inequality) For every µ ∈ X(Ω), there exists a sequence Within the core-radius approach, we carry out the Γ-convergence analysis for more general quadratic functionals than the one in the leading term of (0.2). Specifically, let f : R 2 × R 2×2 → [0, +∞) be a Carathéodory function satisfying the following assumptions: We describe the asymptotic behavior in the logarithmic regime of the functionals where Like the Jacobians in the GL theory, µ is the relevant parameter to keep track of energy concentration.Therefore, we let the functional depend only on µ by setting F ε,δε (µ; w) .
We prove that, for δ ε ε, the functional F ε asymptotically behaves as GL ε , namely, the homogenization process takes place "before" the concentration effect.This implies that the effective cost of a singularity depends on the homogenized energy associated to the functionals F δ (•; E) defined for any open set E as (0.11) Notice indeed that (0.12) In [8] it has been proved that, as δ → 0, the functionals F δ Γ-converge to the homogenized functional F hom (•; E) : H 1 (E; S 1 ) → [0, +∞) defined as In the formula above the energy density Tf hom is the tangential homogenization of the function f (see formula (1.6)).
In order to make the core-radius functionals non-trivial, we define F ε only on the set (0.13) In view of assumption (G) and of the classical results on the core-radius approach functional (see for instance [5, Theorem 3.2]), we have that the functionals F ε satisfy compactness properties analogous to the ones established in Theorem 0.1.
The following theorem on the asymptotic limit of the functionals F ε is proved in Section 6.1.
(ii) ( Γ-limsup inequality) For every µ ∈ X(Ω), there exists a sequence {µ ε } ε ⊂ X(Ω) with In the statement above F 0 : X(Ω) → [0, +∞) is the functional defined as where Ψ(z ; Tf hom ), introduced in (2.9), is the asymptotic energy cost of a singularity of degree z in a homogeneous medium whose energy is F hom .The function Ψ(z ; Tf hom ) is obtained via an asymptotic cell-problem formula and a relaxation procedure.Loosely speaking, we first introduce the minimal F hom energy in an annulus around a singularity with degree z and we show that such a quantity admits a finite limit, denoted by ψ(z; Tf hom ), when the quotient of the radii goes to +∞ .Then Ψ(•; Tf hom ) is obtained as the relaxation of the function ψ(• ; Tf hom ) on Z (see formula (2.9)), accounting for the fact that a singularity of degree z can be approximated by a family of singularities of degree z j with j z j = z .In the simple case that f 3).Using a scaling argument, in Proposition 3.2 we show that ψ(•; Tf hom ) is also the asymptotically minimal F ε energy on "fat" annuli around a vortex of degree z.Here, "fat" stays for thick enough to contain infinitely many δ ε -periodicity cells.Such a property allows to apply the homogenization result in [8] that we show to hold even if the functionals are subject to a degree constraint (see Theorem 1.5 & Corollary 1.6).
A further technical aspect of our analysis is the use, in the proof of the lower bound, of a refinement of the celebrated ball construction introduced in [43,33].This method allows to find a one-parameter family B ε (t) of growing and merging balls, that in turn identify a family of annuli where the energy concentrates.In our case, using a strategy similar to [25], we stop the process at an appropriate "time" t ε at which the constructed family of annuli is "fat" enough to apply the analysis described above and to obtain the desired lower bound.
Using the same strategy exploited for δ ε ε we are able to study the asymptotic behavior of the core-radius approach and of the Ginzburg-Landau functionals also for δ ε ≫ ε (see also [30] for an example in this case).More precisely, we assume that δ ε → 0 as ε → 0 and that (0.15) which implies that lim ε→0 δε ε = +∞ .Furthermore, we assume that (0.16) for some measurable (0, 1) 2 -periodic function a with a(y) ∈ [α, β] ⊂ (0, +∞) for a.e.y ∈ R 2 .The main results in this scaling regime are the following two theorems proven in Section 7.
(ii) ( Γ-limsup inequality) For every µ ∈ X(Ω), there exists a sequence We conclude the introduction with a few comments and remarks about perspectives.A natural follow-up of our results is the extension of our analysis to GL energies with more general integrands in the leading term as those considered in the core-radius approach.A necessary first step in this direction is the proof of a homogenization result for energies defined on maps taking values in a tubular neighborhood of S 1 .More specifically, one could relax the S 1 -constraint in the functionals F δ (•; E) in (0.11), assuming the latter to be defined on H 1 (E; B 1+τ \B 1−τ ) for some τ ∈ (0, 1), and then study their asymptotic behavior when both δ and τ tend to 0 .Another possible extension of our model is the analysis of the case of energy density f satisfying mild coercivity assumptions.This would allow to analyze for instance the problem of topological singularities in presence of soft inclusions of the inhomogeneous material.In this respect, an analysis on the behavior of minimizers of GL functionals in perforated domains has been carried out in [12].Another challenging issue is to look at a higher-order description of the functionals GL ε and F ε , that in the homogeneous case leads to the so-called renormalized energy governing the dynamics of the singularities (see for instance [44] for the GL theory and [4] for discrete models exhibiting topological singularities).In our case of vanishing inhomogeneities we expect the corresponding renormalized energy to depend on the functional Tf hom .Furthermore, we believe that some of the techniques developed in this paper can also be used to make progress in studying stochastic homogenization problems in concentration theory, as for instance those in which the energy density f is replaced by a stationary random potential.
We finally note that inhomogeneities in the GL theory can also be introduced in the potential term; e.g., considering energies of the form (0.17) For some homogenization results for energies (0.17) see [10,11,28,29] and the references therein.The results obtained in those papers differ from ours, since the energy in (0.17) describes a different physical system, namely Type II superconductors in presence of small impurities.Note that a complete study of energies of the form (0.17) may require a very complex multi-scale analysis even in the scalar case (see e.g.[27,20,22]).
Acknowledgements: The hospitality of the Scuola Internazionale Superiore di Studi Avanzati (SISSA) where part of this research was done is gratefully acknowledged.R. Alicandro, A. Braides, and L. De Luca are members of the Gruppo Nazionale per l'Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). A. Braides acknowledges the MIUR Excellence Department Project awarded to the Department of Mathematics, University of Rome Tor Vergata, CUP E83C18000100006.M. Cicalese was supported by the DFG Collaborative Research Center TRR 109, "Discretization in Geometry and Dynamics".

Notation and preliminary results
Basic notation.Given two vectors x, y ∈ R 2 , x • y denotes their scalar product.As usual, the norm of x is denoted by |x| = √ x • x.For every r > 0 and x ∈ R 2 , B r (x) denotes the open ball of radius r centered at x.For x = 0 we also write B r in place of B r (0).S 1 denotes the boundary of B 1 , namely the unit circle in R 2 .Given a ∈ R, ⌊a⌋ := max{z ∈ Z : z ≤ a} and ⌈a⌉ := min{z ∈ Z : z ≥ a} denote the integer parts of a from below and from above, respectively.
The imaginary unit is denoted by ι ∈ C and the complex number e ιa = cos a + ι sin a ∈ C is identified with the Euclidean vector (cos a, sin a) ∈ R 2 .The identification extends to all S 1 -valued maps that can be intended as complex functions as well, if needed.In particular, for every z ∈ Z, by (x/|x|) z we mean the complex function obtained by taking the z-th complex power of the function x/|x|.We say that a family {g η } η converges to g 0 as η → 0 in the topology T , and we write g η T −→ g 0 whenever g ηn T −→ g 0 for any null sequence {η n } n∈N .With a little abuse of terminology the family {g η } η is still called a sequence.The letter C denotes a positive constant whose value may change each time we write it.
Weak star and flat convergence.Let Ω ⊂ R 2 be an open and bounded set with Lipschitz boundary.C c (Ω) denotes the space of continuous functions compactly supported in Ω endowed with the supremum norm.We say that a sequence {µ n } n∈N of measures converges weakly star in Ω to a measure µ, and we write as n → +∞ .
C 0,1 (Ω) denotes the space of Lipschitz continuous functions on Ω endowed with the following norm and we let C 0,1 c (Ω) be its subspace of functions with compact support.The norm in the dual of C 0,1 c (Ω) will be denoted by • flat and referred to as flat norm, while flat → denotes the convergence with respect to this norm.
Notice that Jv can be written in a divergence form as Equivalently, we have Jv = curl (v 1 ∇v 2 ) and Jv = 1 2 curl j(v), where is the so-called current associated to v.
Let A ⊂ Ω be an open set with Lipschitz boundary, and let h ∈ H where τ is the tangent field to ∂A and the product in the above formula is understood in the sense of the duality between H 1 2 and H − 1 2 .In [16,21] it is proven that the definition above is well-posed, it is stable with respect to the strong convergence in here and in what follows we identify v with its trace).Finally, if v ∈ H 1 (A; R 2 ) and |v| = 1 on ∂A, by Stokes' theorem (and by approximating v with smooth functions) one has that (1.2) Note that any v ∈ H 1 (A; R 2 \ B c ) can be written in polar coordinates as v(x) = ρ(x)e ιu(x) on ∂A with |ρ| ≥ c.The function u is said to be a lifting of v.By [14] (see also [15,Theorem 3 and Remark 3]), if A is simply connected, then deg(v, ∂A) = 0 and the lifting can be selected in H 1 2 (∂A) with the map v → u being continuous.For A not necessarily simply connected, if Γ is a connected component of ∂A and the degree of v on Γ is equal to z ∈ Z, then the lifting jumps on Γ by 2πz, but it can be locally selected to belong to be the annulus of radii r and R centered at ξ, and let v ∈ H 1 (A r,R (ξ); S 1 ).Then for every cut L such that A r,R (ξ) \ L is a simply connected set, there exists a lifting u ∈ H We introduce a notion of modified Jacobian (a variant of the notion introduced in [1]), which we will use in our Γ-convergence results.Given 0 < ζ < 1 we define for ρ ∈ [0, +∞) the function Note that, for every v := (v 1 , v 2 ) and w := (w 1 , w 2 ) belonging to .
Lemma 1.1.There exists a universal constant C > 0 such that for any v, w As a corollary of Lemma 1.1 we obtain the following proposition.
Periodic homogenization of energies defined on S 1 -valued maps.In the following paragraph we state some useful propositions regarding the periodic homogenization of energy functionals defined on maps from R 2 to S 1 .The propositions below have been proven in [8] in the more general case of manifold-valued maps defined on R d with d ∈ N. We specialize them here in the S 1 -version that we exploit in the following sections.Let f : R 2 × R 2×2 → [0, +∞) be a Carathéodory function satisfying assumptions (P) and (G).For every δ > 0 and for every open bounded set E ⊂ R 2 we define the functional For every s = (s 1 , s 2 ) ∈ S 1 we set s ⊥ = (−s 2 , s 1 ) and T s (S 1 ) = Rs ⊥ = {λs ⊥ : λ ∈ R} denotes the tangent space of S 1 at the point s.We also introduce the set and for every (s, M ) = (s, s ⊥ ⊗ ξ) ∈ T S 1 we define (1.6) where Q := (0, 1) 2 .The function Tf hom is called the tangential homogenization of the function f .The function Tf hom is a tangentially quasi-convex function according to the following definition.We say that a Borel function h : We note that the function Tf hom satisfies the following property: We define the functional The following theorem has been proven in [8,Theorem 1.1].
For every 0 < r < R and for every x ∈ R 2 we set A r,R (x) := B R (x)\B r (x) and A r,R := A r,R (0).Moreover, for every z ∈ Z \ {0} we define Given z ∈ Z\{0}, for every δ > 0 we define the functionals and The next result is a consequence of Theorem 1.3.
for some w 0 ∈ H 1 (A r,R ; S 1 ) .By standard Fubini arguments, for almost every r < ρ < R, we have that the trace of w δ on ∂B ρ is bounded in H 1 (∂B ρ ; S 1 ) and hence (up to a not relabeled subsequence) it weakly converges to a function g ρ .Since w δ − w 0 L 2 (Ar,R;R 2 ) → 0 , we get that g ρ = w 0 for a.e.ρ ∈ (r, R).By the very definition of degree in (1.2), deg(w 0 , ∂B r ) = z and hence w 0 ∈ A r,R (z).
The following corollary holds true as a consequence of (G), (1.8), Theorem 1.5 and thanks to the well-known property of convergence of minima in Γ-convergence (see [17,18,23]).

The effective energy of a singularity
In this section we introduce and discuss the properties of the minimal energy cost Ψ(z; h) of a vortex like singularity of degree z for a homogeneous quadratic functional of energy density h defined on S 1 -valued maps.The function Ψ(•; h) is crucial in order to determine the Γ-limits for both the cases δ ε ε and δ ε ≫ ε , choosing h = Tf hom , with Tf hom defined in (1.6).On the one hand (see Section 6) for δ ε ε , Ψ(z; Tf hom ) turns out to be the effective energy cost of a singularity of degree z (see Theorems 6.1 and 6.2).On the other hand (see Section 7) for δ ε ≫ ε , recalling the definition of λ in (0.15), we have that λΨ(z; Tf hom ) is the effective energy cost of a singularity of degree z on scales of order between δ ε and 1 (see Theorems 7.1 and 7.2) .
Let h : T S 1 → [0, +∞) be a continuous function, tangentially quasi-convex according to (1.7), and such that Assume moreover that there exist α, β such that 0 < α ≤ β and For every open bounded set E ⊂ R 2 we define the functional H(•; E) : H 1 (E; S 1 ) → [0, +∞) as Given z ∈ Z \ {0} and 0 < r < R we set Making the change of variable y = x r and considering the 2-homogeneity (2.1) of the function h we conclude that, for every w ∈ H 1 (A r,R ; S 1 ), the following relation holds: where ŵ(y) := w(ry).Gathering together (2.4) and (2.3) we deduce that

2) and (2.1).
For z ∈ Z \ {0} and 0 < r < R let ψ r,R (z, h) be the function defined in (2.3).Then there exists the limit Proof.In view of (2.5), it is enough to prove the inequality Denoting by w R a minimizer of (2.3), letting k = kR,ρ ∈ {1, . . ., K R,ρ } be such that and setting ŵρ, k(y) := w R (ρ k−1 y), we obtain min where the last equality follows by (2.1).By the very definition of K R,ρ we conclude that The inequality above yields (2.7) on taking first the limit as R → +∞ and then as ρ → +∞.
In the next proposition we show that if f is of the form in (1.10), then Ψ(z; Tf hom ) equals |z| up to a constant pre-factor.
where A hom is defined in (0.4).
Then the domain A r,R \ L is simply connected.We set z)}.By Remark 1.4 and by (2.3) we have (2.12) where the last equality follows from the fact that A hom is symmetric and hence √ A hom is.Setting û(y) := u( √ A hom y), we have that ∇û(y) := √ A hom ∇u( √ A hom y).Thus the change of variables x = √ A hom y in (2.12) yields where we have set For sufficiently large R/r there exist 0 < λ < Λ that depend only on A hom and do not depend on r and R such that A λr,ΛR ⊂ ( √ A hom ) −1 (A r,R ) so that, by (2.13), , whence (2.11) follows using the very definition of Ψ in (2.9).

Asymptotic analysis on annuli
In this section we prove some auxiliary results on the asymptotic behavior of the minimal energy on an annulus when its inner and outer radii are powers of ε.Such results will be crucial in the proofs of the Γ-convergence theorems in Section 6 and in Section 7. The next lemma states that the minimum in (2.3) for H = F δ changes by at most a multiple of z 2 if the competitors are chosen with fixed trace (x/|x|) z instead of fixed degree z, thus belonging to a new appropriate set of admissible functions defined as and with f satisfying condition (G).Then, there exists a constant C = C(α, β) > 0 such that, for every z ∈ Z \ {0}, Proof.The first inequality in (3.2) follows from the inclusion A r,R (z) ⊂ A r,R (z).Hence, it is enough to show that for every w ∈ A r,R (z) there exists w ∈ A r,R (z) such that for some constant C depending only on the constants α and β in (G).Set K := ⌊ log R−log r log 2 ⌋ and We first consider the case that there is at most one annulus A k such that Then, in view of (G), we have whence, using also (3.4), we deduce From now on, we can assume that (3.5) is satisfied by at least two of the annuli A k .We let k 1 and k 2 denote the smallest and the largest k ∈ {1, . . ., K} satisfying (3.5).Let moreover L := {(0, x 2 ) : −R ≤ x 2 ≤ −r} be a cut of the annulus A r,R such that the domain A r,R \ L is simply connected.By [14], there exists a lifting u ∈ H we have that the function u jumps by 2πz across L. By the properties of the lifting, Furthermore, setting ) is a lifting of ( x |x| ) z .Using the complex notation we set w := e ι u , where the lifting u is defined as In the formula above, for i = 1, 2 the function . By construction and by (3.4), we have that Therefore, in view of (3.8) it is enough to prove that the energy of w on A k1 and A k2 is bounded from above by C|z| 2 for some constant C > 0 depending only on α and β.We prove this fact only for the annulus A k1 , being the proof for A k2 similar.To this end, we notice that in A k1 one has that (3.9) where the second inequality is a consequence of the Poincaré-Wirtinger inequality applied to the domain A k1 \ L, and the third inequality follows on gathering together (3.5), (3.6), and (G).Note that all the constants appearing in (3.10) depend only on α and β.By integrating (3.9) and using (3.10), (3.5), (3.6) and (G), we deduce that thus concluding the proof of (3.3).
In the next proposition we show that in the | log ε| regime, to some extent, the homogenization process commutes with the minimization process defining ψ(d; Tf hom ).Proposition 3.2.Let F δε be defined in (0.11) with f satisfying assumptions (P), (G), (H), and let Tf hom be defined in (1.6).Then for any s 1 and s 2 such that 0 ≤ s 1 < s 2 < 1 and lim ε→0 δε ε s 2 = 0 we have (3.11)lim where ψ(z; Tf hom ) is the function defined in (2.6) with h = Tf hom .
Proof.We first show that (3.12) lim inf To this purpose, we fix for some constant C (independent of ε) and let k = kε,R ∈ {1, . . ., K ε,R } be such that By the change of variable y = and by property (H), we have Therefore, since by assumption lim sup ε→0 δε R k ε s 2 = 0 for every k = 1, . . ., K ε,R , by using (3.13), (3.14), (3.15), and Corollary 1.6, we deduce that lim inf Formula (3.12) follows from the estimate above as R → +∞ thanks to Proposition 2.1 applied to h = Tf hom .To conclude the proof of (3.11) we are left to show that To this purpose, we take R > 1 and set J ε,R := ⌈(s 2 − s 1 ) | log ε| log R ⌉ .We observe that (3.17) inf We also note that for every R > 1, thanks to Corollary 1.6, there exists a modulus of continuity ω such that inf F δ (w; A 1,R ) ≤ min F hom (w; A 1,R ) + ω(δ).
Remark 3.3.Note that (3.11) holds true also if the center of the annulus is a point ξ ε depending on ε , since all the estimates in the previous proof do not depend on the center of the annulus.

The ball construction
In this section we present the so-called ball construction introduced in [33,43], which provides lower bounds of the Dirichlet energy in the presence of topological singularities.We slightly revisit it, following the approach by Sandier [43] and adopting the notation in [26] (see also [5]).
Let B = {B r1 (x 1 ), . . ., B rn (x n )} be a finite family of open balls in R 2 with disjoint closure Bri (x i ) ∩ Brj (x j ) = ∅ for i = j and let µ = Let moreover E(B, µ, •) be an increasing set-function defined on open subsets of R 2 satisfying the following properties: for some constant α > 0 .(1) B(0) = B ; (2) U (t 1 ) ⊂ U (t 2 ) for any 0 ≤ t 1 < t 2 ; (3) the balls in B(t) are pairwise disjoint; (4) for any 0 ≤ t 1 < t 2 and for any open set Proof.In order to construct the family B(t), we closely follow the strategy of Sandier and Jerrard in [33,43].It consists in letting the balls alternatively expand and merge into each other as follows.
In the expansion phase the balls expand, without changing their centers, in such a way that, at each (artificial) time t the radius r i (t) of the ball centered at x i satisfies The first expansion phase stops at the first time T 1 when two balls bump into each other.Then the merging phase begins.It consists in identifying a suitable partition {S 1 j } j=1,...,Nn of the family B ri(T1) (x i ) , and, for each subclass S 1 j , in finding a ball B r 1 j (x 1 j ) which contains all the balls in S 1 j such that the following properties hold: After the merging phase another expansion phase begins: we let the balls B r 1 j (x 1 j ) expand in such a way that, for t ≥ T 1 , for every j we have that (4.4) r 1 j (t) Again note that r 1 j (T 1 ) = r 1 j .We iterate this procedure thus obtaining a discrete set of merging times {T 1 , . . ., T K } with K ≤ n and a family B(t) for all t ≥ 0.More precisely, B(t) is given by {B rj(t) (x j )} j for t ∈ [0, T 1 ); for t ∈ [T k , T k+1 ), B(t) can be written as {B r k j (t) (x k j )} j for all k = 1, . . ., K − 1, while it consists of a single expanding ball for t ≥ T K .By construction, we clearly have properties (1), ( 2) and (3).Moreover, ( 5) is an easy consequence of (4.3), (4.4) and property P2).
It remains to show property (4).We preliminarily note that, by (2), for every open set U ⊂ R 2 (4.5) Let t 1 < t < t 2 .In view of (4.5) and since E is an increasing set-function satisfying property (i), if we show that (4) holds true for the pairs (t 1 , t) and ( t, t 2 ), then (4) follows also for t 1 and t 2 .Therefore, we can assume without loss of generality that T k / ∈]t 1 , t 2 [ for any k = 1, . . ., K. Let t 1 < τ < t 2 and let B ∈ B(τ ).Then there exists a unique ball B ′ ∈ B(t 1 ) such that B ′ ⊂ B. By construction, µ(B) = µ(B ′ ) and by (4.1) we have that which, summing up over all B ∈ B(τ ) with B ⊂ U , and using (4.5), yields Property (4) follows by letting τ → t 2 .
We recall the following well-known lemma (see e.g., [26, Lemma 2.2]) for the reader's convenience.Then, there exists a constant C > 0 such that

General Γ-liminf inequality
In this section we state and prove an asymptotic lower-bound estimate for general core-radius approach functionals (see Propositions 5.2 and 5.4); such results will be instrumental for the proofs of the Γ-liminf inequalities in Theorems 0.2, 0.4, 0.5 and 0.6.
We introduce the increasing set-function E satisfying the assumptions (i) and (ii) in Section 4 as follows.Let B = {B r1 (x 1 ), . . ., B rN (x n )} be a finite family of open balls in R 2 with Bri (x i ) ∩ Brj (x j ) = ∅ for i = j, and let µ = where the supremum is taken over all finite families of disjoint annuli A j ⊂ A that do not intersect any B ri (x i ).Note that, if A is an annulus that does not intersect any B ri (x i ), then E(B, µ, A) = G(B, µ, A).
Remark 5.1.The convenience of introducing E in (5.2) to prove a lower-bound inequality for (an appropriate scaling of) the functional F δε in (0.11) will be clear in the following sections.However, the following simple observation already points in the right direction.Let Ω(B) = Ω \ B∈B B, w ∈ H 1 (Ω(B); S 1 ) and µ := B∈C deg(w, ∂B)δ xB where C denotes the family of balls in B that are contained in Ω, and x B is the center of B. Then, by Jensen's inequality and by the lower bound in (G), we deduce that For every µ ∈ X(Ω) and for every family of pairwise disjoint balls B such that supp µ ⊂ B∈B B, we set AF (µ, B) := {w ∈ H 1 (Ω(B); S 1 ) : deg(w, ∂B) = µ(B) for every B ∈ B}.In addition we set (5.4) where F δε is defined in (0.11) and f satisfies (P), (G), (H) .
..,n , we have that AF (µ, B ε ) coincides with the set AF ε (µ) defined in (0.10) and that F ε (µ, B ε ) = F ε (µ) .We are now in a position to state the first main result of this section, concerning the case δ ε ε .

Now we extend the function
) in such a way that for every B and B as above for some universal constant Ĉ.We consider B = B R (ξ) and B = B cR (ξ) two balls as above.Since deg(w ε , ∂B R (ξ)) = deg(w ε , ∂B cR (ξ)) = 0, by arguing as in [14] (see also [15]), one can show that there exists a lifting u . We define the function û By the Poincaré-Wirtinger inequality and by the very definition of U ε we have that there exists a constant Ĉ (independent of ε) such that ε ( tε,1) B; S 1 ) and satisfies (5.14).Then from (5.12) and (5.14) we deduce that (5.15) We now focus on the balls in C =0 ε ( tε,1 ) .We set µ( tε,1 ) := B∈C =0 ε ( tε,1) µ ε (B)δ xB .In view of the ball construction in Section 4 and of (5.10), we have that ♯C =0 ε ( tε,1 ) ≤ C p .Therefore, up to extracting a subsequence we may assume that ♯C =0 ε ( tε,1 ) = L for every ε > 0 and for some L ∈ N .For every l = 1, . . ., L, let x l ε be the center of the l-th ball B l ε in C =0 ε ( tε,1 ) .Up to a further subsequence, we can assume that the points x l ε converge to some points in the finite set {ξ 0 = x 0 , ξ 1 , . . ., ξ L ′ } ⊂ Ω, where L ′ ≤ L .Let ρ > 0 be such that B 2ρ (x 0 ) ⊂⊂ Ω and B 2ρ (ξ j ) ∩ B 2ρ (ξ k ) = ∅ for all j = k .Then x l ε ∈ B ρ (ξ j ) for some j = 1, . . ., L ′ and for ε small enough.We set με := By construction, we have that (5.16) which, in view of (5.10) and (5.11), implies that, up to a subsequence, με * ⇀ µ = z 0 δ x0 .Therefore, for sufficiently small ε, Thanks to (5.15) and the assumption (G), we have that It remains to prove the lower bound for the right-hand side of (5.18) .To this end, we take 0 < p ′ < p such that R ε ( tε,1 ) ≤ ε p ′ (note that such p ′ always exists since, by Lemma 5.3, ), choose 0 < p < p ′ and let g ε : [p, p ′ ] → {1, . . ., L} denote the function which associates to any q ∈ [p, p ′ ] the number g ε (q) of connected components of the set L l=1 B ε q (x l ε ) .For every ε > 0 , the function g ε is monotonically non decreasing so that it can have at most L ≤ L discontinuities.Let q j ε , for j = 1, . . ., L, denote the discontinuity points of g ε and assume that p There exists a finite set △ = {q 1 , q 2 , . . ., q L} with q i < q i+1 and L ≤ L such that, up to a subsequence, {q j ε } ε converges to some point in △ , as ε → 0 for every j = 1, . . ., L .Without loss of generality we may assume that q 1 = p , and that q L = p ′ .Let λ > 0 be such that 4λ < min{q i+1 − q i : i ∈ {1, 2, . . ., L}} and let ε be so small that for every j = 1, . . ., L , |q j ε − q i | < λ for some q i ∈ △.Then the function g ε is constant in the interval [q i + λ, q i+1 − λ] , its value being denoted by M i ε .For every i = 1, . . ., L − 1 we construct a family of ) and m = 1, . . ., M i ε .The annuli C i,m ε can be taken pairwise disjoint for all i and m and such that for all i = 1, . . ., L − 1 .Note that, for ε small enough, C i,m ε ⊂ B 2ρ (x 0 ) for all i and m .By (5.16) we have that |µ ε (B ε q i+1 −λ (y m ε ))| ≤ C for every i = 1, . . ., L − 1 and m = 1, . . ., M i ε .Therefore, up to passing to a further subsequence, we can assume that , with M i and z i,m independent of ε .Finally, in view of (5.17), we have that (5.19) Observe that the assumption lim sup ε→0 δε ε < +∞ implies the inequality lim ε→0 δε ε q i+1 −λ = 0 for every i.Hence, we can apply Proposition 3.2 with s 1 = q i + λ < q i+1 − λ = s 2 (see also Remark 3.3) to get that for every i and m there exists a modulus of continuity ω such that Summing the previous inequality over m and i and using (5.18) yields (5.20) where the second inequality follows from (5.19) and from the very definition of Ψ in (2.9).Then, the claim follows by (5.20) taking the limits as ε → 0 , λ → 0 , p → 0 , and p, p ′ → 1 and using (5.7).
As for the case δ ε ≫ ε , we restrict our analysis to functionals of the form (0.16).In such a case the main result is the following.Proposition 5.4.Let F δε , F ε be defined in (0.16), (5.4), respectively, where a is a measurable Rad(B ε ) ≤ Cε| log ε|.
where A hom is defined in (0.4).
Proof.The proof closely resembles the one of Proposition 5.2; here we only highlight the main changes that are needed to prove the different lower bound in the regime (7.1). Let for some constant C independent of ε.By a standard localization argument in Γ-convergence, we can assume that µ = z 0 δ x0 for some z 0 ∈ Z \ {0} and x 0 ∈ Ω .For every ε > 0, let B ε (t) be a time-parametrized family of balls introduced as in Proposition 4.2, starting from B ε =: B ε (0) .For every t ≥ 0, we set R ε (t Fix λ < p < 1 .By arguing as in the proof of (5.10) and (5.11), we have that Following the reasoning in the proof of Proposition 5.2 we have that for every 0 < η < p − λ there exists t ε (p) ≤ tε,1 ≤ t ε (p − η) and a a map ŵε ∈ H 1 (Ω(B ε ) ∪ B∈C =0 ε ( tε,1) ; S 1 ) (with C =0 ε ( tε,1 ) defined in (5.13)) satisfying (5.15).

The case δ ε ε
This section is devoted to the proofs of Theorems 0.4 and 0.2.
6.1.The core-radius approach.For the reader's convenience, we re-state Theorem 0.4 and recall that X ε (Ω) is defined in (0.13).
In view of (6.1), by applying (5.3) with U = Ω, we have that (6.2) where E is defined in (5.1)-(5.2) .By (6.2) and the Jensen inequality, considering the definition of X ε (Ω), we get whence we deduce that (6.4) The claim follows by Proposition 5.2 whose assumptions are fulfilled in view of (6.3) and (6.4).

6.2.
The Ginzburg-Landau model.This subsection is devoted to the proof of Theorem 0.2, which we prove here under slightly more general assumptions on the potential term.More specifically, we consider and, we define We can re-state Theorem 0.2 as follows.
(iii) ( Γ-limsup inequality) For every µ ∈ X(Ω), there exists a sequence such that Jv ε flat → πµ and Proof.Since a ≤ β a.e., the compactness property (i) is a corollary of classical results in the variational analysis of the classical GL functional (see for instance [5,Theorem 4.1]).

The case δ ε ≫ ε
This section is devoted to the proofs of Theorems 0.5 and of 0.6.We will prove the above Γ-convergence results under the assumption that (7.1) lim 7.1.The core-radius approach.For the reader's convenience, we re-state Theorem 0.5 and we recall that X ε (Ω) is defined in (0.13).

A
hom ∇u, ∇u dx : u ∈ H 1 (B R \ B r ), deg (e ıu ; B r ) = z , from which the Γ-limit is obtained by locally optimizing the degree (possibly approximating a vortex by more vortices).A computation eventually allows to conclude that the limit energy has the

|∇u| 2
dx : u ∈ H 1 (B R \ B r ), deg (e ıu ; B r ) = z ,for some choice of r and R with R/r → +∞, and η → 0. This can be done by a scaling argument if δ ≪ ε.An approximation argument, more in general, allows to extend this result to (0.7)| log δ| ≥ | log ε| using the scaling properties of the energies.

Remark 4 . 1 .
Let w ∈ H 1 loc (R 2 \ B∈B B; S 1 ) be such that µ = B∈B deg(w, ∂B)δ xB , where x B is the center of B. Then, an explicit example of admissible functional E(B, µ, •) is given by E(B, µ, A) := α A\ B∈B B |∇w| 2 dx, for every open set A ⊂ R 2 .For further details see Remark 5.1.For every ball B ⊂ R 2 , let r(B) denote the radius of the ball B; moreover, for every family B of balls in R 2 we set Rad(B) := B∈B r(B).

Proposition 4 . 2 .
There exists a one-parameter family of open balls B(t) with t ≥ 0 such that, setting U (t) := B∈B(t) B, the following conditions are fulfilled:

Lemma 4 . 3 .
Let B be a family of pairwise disjoint balls in R 2 and let C be the family of balls in B which are contained in Ω.Let moreover ν 1 , ν 2 be two Radon measures supported in Ω with supp ν 1 ⊂ B∈C B, supp ν 2 ⊂ B∈B B and ν 1 (B) = ν 2 (B) for any B ∈ C .