Ginzburg-Landau model with small pinning domains

We consider a Ginzburg-Landau type energy with a piecewise constant pinning term $a$ in the potential $(a^2 - |u|^2)^2$. The function $a$ is different from 1 only on finitely many disjoint domains, called the {\it pinning domains}. These pinning domains model small impurities in a homogeneous superconductor and shrink to single points in the limit $\v\to0$; here, $\v$ is the inverse of the Ginzburg-Landau parameter. We study the energy minimization in a smooth simply connected domain $\Omega \subset \mathbb{C}$ with Dirichlet boundary condition $g$ on $\d \O$, with topological degree ${\rm deg}_{\d \O} (g) = d>0$. Our main result is that, for small $\v$, minimizers have $d$ distinct zeros (vortices) which are inside the pinning domains and they have a degree equal to 1. The question of finding the locations of the pinning domains with vortices is reduced to a discrete minimization problem for a finite-dimensional functional of renormalized energy. We also find the position of the vortices inside the pinning domains and show that, asymptotically, this position is determined by {\it local renormalized energy} which does not depend on the external boundary conditions.


Introduction and main results
In this work we study the minimizers of the Ginzburg-Landau type functional where Ω ⊂ C is a bounded, smooth, simply connected domain, ε is a positive parameter (the inverse of the Ginzburg-Landau parameter κ = 1/ε), δ = δ(ε) > 0 is a geometric parameter and u is a complex-valued map. In order to define the function a δ , we need to introduce the notion of a pinning domain.
We now define a δ : Ω → {b, 1}, b ∈ (0, 1) as: The functionals of this type arise in models of superconductivity for composite superconductors. The experimental pictures suggest nearly 2D structure of parallel vortex tubes ( [25], Fig I.4). Therefore, the domain Ω can be viewed as a cross-section of a multifilamentary wire with a number of thin superconducting filaments. Such multifilamentary wires are widely used in industry, including magnetic energy-storing devices, transformers and power generators [17], [16].
Another important practical issue in modeling superconductivity is to decrease the energy dissipation in superconductors. Here, the dissipation occurs due to currents associated with the motion of vortices ( [21], [6]). This dissipation as well the thermomagnetic stability can be improved by pinning ("fixing the positions") of vortices. This, in turn, can be done by introducing impurities or inclusions in the superconductor. In the functional (1) the set ω δ models the set of small impurities in a homogeneous superconductor. The size of the impurities in our model is characterized by the geometric parameter δ which goes to zero together with the material parameter ε. We assume henceforth that | ln δ(ε)| 3 | ln ε| → 0. (H) Essentially, this condition means that δ is much larger than ε on the logarithmic scale. For example, if ε = 2 −j and δ(ε) = 2 −k(j) , then (H) implies that k(j) 3 j → 0.
Notation. In what follows: • We consider a sequence ε n ↓ 0 and we write ε instead of ε n ; the dependence of ε on n is implicit.
Here "×" stands for the vectorial product in C, i.e. z 1 × z 2 = Im(z 1 z 2 ), z 1 , z 2 ∈ C, and ∂ τ is the tangential derivative. The degree is an integer, and the condition deg ∂Ω (u) = d > 0, u ∈ H 1 (Ω, C) implies that u must have at least d zeros (counting multiplicity) inside Ω. The properties of the topological degree can be found, e.g., in [13] or [8].
Minimization problems for Ginzburg-Landau type functionals have been extensively studied by a variety of authors. The pioneering work on modeling Ginzburg-Landau vortices is the work of Bethuel, Brezis and Hélein [11]. In this work the authors suggested to consider a simplified Ginzburg-Landau model (1) with a ≡ 1 in Ω (i.e. without pinning term), in which the physical source of vortices, the external magnetic field, is modeled via a Dirichlet boundary condition with a positive degree on the boundary (3). The analysis of full Ginzburg-Landau functional, with induced and applied magnetic fields, was later performed by Sandier and Serfaty in [27].
The functional (1) with non-constant a(x) was proposed by Rubinstein in [26] as a model of pinning vortices for Ginzburg-Landau minimizers. Shortly after, André and Shafrir [4] studied the asymptotics of minimizers for a smooth (say C 1 ) a. One of the first works to consider a discontinuous pinning term, which models a composite two-phase superconductor, was [18]. In this work, a single inclusion described by a pinning term independent of the parameter ε was considered for a simplified Ginzburg-Landau functional with Dirichlet boundary condition g on ∂Ω. Namely the pinning term is here ω is a simply connected open set s.t. ω ⊂ Ω. The main objective of [18] was to establish that the vortices are attracted (pinned) by the inclusion ω, and their location inside ω can be obtained via minimization of certain finite-dimensional functional of renormalized energy. Full Ginzburg-Landau model with discontinuous pinning term was later considered by Aydi and Kachmar [5]. An ε-dependent but continuous pinning term a ε (x) was studied by Aftalion, Sandier and Serfaty in [1]. The work [3] studies the case of the smooth a with finite number of isolated zeros, and in [2] the pinning term a takes negative values in some regions of the domain Ω. The other works related to Ginzburg-Landau functional with pinning term include, e.g., [21], [28].
In this work, we consider the minimization problem (1)-(3) with a discontinuous pinning term given by (2). We prove that despite the fact that a ε → 1 a.e. as ε → 0, i.e. the pinning term disappears in the limit, the pinning domains ω δ capture the vortices of Ginzburg-Landau minimizers of (1) for small ε.
The main difficulty in the analysis of this problem stems in the fact that the a priori Pohozhaev type estimate 1 − |v| 2 2 L 2 (Ω) ≤ Cε 2 for the minimizer v (on which the analysis in [11] and [18] is based) does not hold. Therefore, we develop a different strategy of reducing the study of the minimizers of (1) to the analysis of S 1 -valued maps via the uniform estimates on the modulus of minimizers away from the pinning domains (see Proposition 5 below).
Following [18], let U ε be the unique global minimizer of Using the Substitution Lemma of [18], we have that for v ∈ H 1 g , From the decomposition (4), we can reduce the minimization problem (1)-(3) to the minimization problem for F ε in H 1 g , namely, the minimizer v ε of F ε in H 1 g has the same vorticity structure as the original minimizer u ε of (1)- (3).
Depending on the relation between M (number of inclusions), and d (number of vortices), we distinguish two cases: Case I: M ≥ d (more inclusions than vortices), Case II: M < d (more vortices than inclusions).
For example, we are going to show that for the minimizer v ε : • if M = 3 and d = 2 (Case I), we have two distinct inclusions containing exactly one zero each, • if M = 2 and d = 3 (Case II), we have one zero inside one inclusion and two distinct zeros inside the other inclusion.
Generally speaking, outside a fixed neighborhood of d ′ = min {d, M } inclusions (centered at a = (a i 1 , ..., a i d ′ )), the minimizer v ε is almost an S 1 -valued map. Moreover, by minimality of v ε , the selection of centers of inclusion containing its zeros and the distribution of degrees of v ε around the a i 's are related to the minimization of the Bethuel-Brezis-Hélein renormalized energy W g . In other words, we reduce the problem of finding vortices of the minimizers v ε to a two-step procedure. As the first step, we determine the inclusions with vortices, which is a discrete minimization problem for W g and is significantly simpler then the minimization of this renormalized energy functional over Ω d ′ . Secondly, we determine the locations of the zeros (vortices) locally inside each inclusion and show that their positions depend only on b, on the geometry of ω and on the relation between d and M , but not on the external Dirichlet boundary condition g (see Theorem 4 below). Our main result in Case I is the following: . For any sequence ε n ↓ 0, possibly after passing to a subsequence, there are d distinct points 2.1 Properties of U ε Proposition 1 (Maximum principle for U ε , [18] Proposition 1). The special solution U ε satisfies b ≤ U ε ≤ 1 in Ω.

Proposition 2.
There are C, c > 0 (independent of ε) s.t. for any R > 0 we have The proof of the Proposition 2 is presented in the Appendix A.

Upper bound in Case I: M ≥ d
There is a constant C depending only on g, ω and Ω s.t. we have

Upper bound in Case II: M < d
There is a constant C depending only on g, ω and Ω s.t. for all The proof of Proposition 3 is given in Appendix B.

Identifying bad discs
Consider the weighted Ginzburg-Landau functional Denote v ε a minimizer of F w ε in H 1 gε . Then the following results hold: 2. Let µ ∈ (0, 1). Then there are ε 0 , C > 0 depending only on b, µ, Ω, Lemma 1 is proved in Appendix C.

A model problem: one inclusion
By combining the results of Section 2, the proofs of both Theorem 1 and Theorem 2 are based on the analysis of two distinct problems: 1. A minimization problem of the Dirichlet functional among S 1 -valued map defined on a perforated domain.
2. The study of the minimizers v ε around an inclusion.
This section focuses on the second problem. More precisely, we fix ρ > 0 and study the minimization problem of F ε (·, B(a i , ρ)) with variable boundary conditions. Fix ρ > 0 and let and Assume also that deg in the class Without loss of generality assume a i = 0. Let v ε be a minimizer of (15) in (16). Performing the change of variablesx = x δ in (15), we have Here, for a map w ∈ H 1 (B(0, ρ)), we denoteŵ(x) := w(δx) and ξ = ε δ . The class (16) under this change of variables becomeŝ Note that the above rescaling enables us to fix the pinning domain independently of ε. The asymptotic behavior ofv ε will be obtained in several steps: • We first establish a bound for |v ε | (Proposition 5). This bound will allow us to localize (roughly) the vortices of v ε near the inclusion.
• We next establish sharp energy estimates (Proposition 6) and use them to obtain the uniform convergence of solutions away from the inclusion (Proposition 7 and Corollary 2). We establish the strong H 1 convergence of solutions away from the "vortices" (Proposition 8) and derive the equation satisfied by the limiting map (Proposition 10).
• The last step is the location and quantization of the vorticity: for small ε, the minimizers admits exactly d 0 zeros, and all the zeros lie in the inclusion and have a degree equal to 1 (Propositions 8 and 11).
In order to obtain a lower bound for F ε we use the identity The first integral in (23) is estimate via (22): By combining (20) and Proposition 2, we have for small ε here we rely on the assumption (H) on the behavior of δ(ε) as ε → 0. Substituting the bounds (24) and (25) in (23) we obtain a contradiction with (11). This completes the proof of Proposition 5.
3.2 Distribution of Energy in B(0, ρ δ ) Proposition 6. The following estimates hold: and Proof. We start by proving thatF As before, we use Theorem 4.1 in [27]: for 0 < r < r 0 := 10 −2 · dist(ω, ∂B(0, 1)), there are C > 0 and a finite covering by disjoint balls B ε 1 , ..., B ε N (with the sum of radii at most r) of the set with D = j |d j | and It follows that D ≥ d 0 and then (28) is a direct consequence of (29) and the boundÛ ε ≥ b.
We next prove that there is C > 0 s.t.
Using exactly the same techniques as in the proof of Proposition 6, one may easily prove the following estimate.
and there is Proof. From Proposition 2 As in [18], the following expansion holds Using (10), we have With (37) and (38), we conclude that E ξ (Û εvε , K) = O(1). SinceÛ ε andÛ εvε satisfy the Ginzburg- On the other hand, using the fact thatv ε is bounded in C k (K) together with the equation ofv ε , we find that 1 − |v ε | 2 ≤ C K ξ 2 in K.
We are now in position to bound the potential part of the energy.
Proof. Note that from Propositions 4 and 6, we find that there is C > 0 s.t.
Card J ≤ N.
Proof. Since each point of Ω is covered by at most 16 discs B( The previous assertion implies that Card The next result is a straightforward variant of Theorem IV.1 in [11]. Lemma 3. Possibly after passing to a subsequence and relabeling I, we may choose J ′ ⊂ J and a constant λ ≥ 1 (independently of ε) s.t.
We will say that, for i ∈ J ′ , B(x i , λε 1/4 ) are separated µ-bad discs. From now on, we work with separated µ-bad discs. Denotex i = x i δ . By Proposition 5 we know that for small ε, we havê Note that for i ∈ J ′ , we have

Convergence in H
We have the following theorem.
Note that since by comparing (52) with (53), we also have In order to prove (45), it suffices to establish the convergence and to use the fact that |v ε | → 1 uniformly.
We next prove that, for some (Once proved, this assertion will imply, via Sobolev embedding that (55) holds.) Note that (up to a subsequence) From Theorem 2 in [22], there is 2 < q ≤ p and C > 0 s.t.
Step 3: We prove the third assertion Consequently, we have By combining (47) and (56), we obtain the existence of C independent of ε and η s.t.
We are now in position to estimate the rate of uniform convergence of |v ε | in a compact set Proof. Due to (36), it is sufficient to establish this result in B(0, 1) \ B(α i , η). Combining Corollary 3 with (44), we obtain thatF 3.6 Information about the limit v 0 Following [11] (Appendix IV, page 152) we have Proposition 9. For all 1 ≤ p < 2 and for any compact K ⊂ R 2 ,v ε is bounded in W 1,p (K).

Uniqueness of zeros
Proposition 11. For ε sufficiently small, the minimizerv ε has exactly d 0 zeros.
Proof. It suffices to prove that for small ε there is a unique zero ofv ε in B(α k , r), k = 1, ..., d 0 , with r defined in the proof of Proposition 10. Sinceŵ ε =v εÛε b , from Proposition 2 and Proposition 10 we see that Using (61) and (62) and arguing as in the alternative proof of Theorem VII.4 in [11] (page 74) we obtain that ∇H k (α k ) = 0.
Finally, we are now in position to obtain, as in Theorem IX.1 [11] (using the main result of [7]), that there is a unique zero ofŵ ε (and, therefore, ofv ε ) in B(α k , r).
The proof of Theorem 3 is complete.
• (Section 4.5) Finally, we make a fundamental observation: the limiting function g 0 = lim tr ∂B 1v ε and the points α obtained form Theorem 3 form a minimal configuration for W 1 (g) + W 2 (β,g). Thus, introducingW we conclude that α minimizesW .
In this section we prove the following theorem.
Remark 2. The renormalized energy in the expansion (64) decouples into the part that depends only on the external boundary conditionsW 0 (f 0 ) and the part that depends only on the location of the vorticesW (α). Since α minimizesW , the external boundary data has no effect on the location of vortices inside the inclusion. This is a drastic difference with the results of [11] and [18], where the Dirichlet boundary data on the external boundary influences the location of the vortices.
One may prove that ψ δ L 2 (A δ ) is bounded and more precisely we have the following result.
Proof. Let (a n ) n∈Z , (b n ) n∈Z ⊂ C be s.t.
Following the same lines as Proposition 13 we obtain and It follows from (68) and (69) that From (70) and (67), we deduce that One of the main ingredients in the study of the renormalized energy is that the Dirichlet condition f min (x) = γ 0

Lower bound
We prove that the upper bound (76) is sharp by constructing the matching lower bound.
The goal of this section is to underline an important property of the points α, namely, that they minimize the quantity inf g 0 ∈C ∞ (∂B 1 ,S 1 )W (·, g 0 ).

Proofs of Theorems 1 and 2
In this section v ε is a minimizer of F ε in H 1 g (Ω, C). We split the proofs of Theorem 1 and 2 in three steps: • (Section 5.1) Using estimates on |v ε |, we first localize the vorticity to the neighborhoods of selected inclusions. Then we find two separate energy expansions in two sub-domains of Ω: away from the selected inclusions and around them.
• (Section 5.2) We study the asymptotic behavior of v ε . We prove that, for small ε, v ε has exactly d zeros of degree 1.
• (Section 5.3) We give an expansion of F ε (v ε ) up to o ε (1) terms and relate the choice of the inclusions with vortices to the renormalized energy of Bethuel, Brezis and Hélein.

Locating bad inclusions
The following result gives a uniform bound on the modulus of minimizers away from the inclusions.
We next obtain the following lower bounds for the energy.
By Theorem 4.1 [27], for r = 10 −2 there are C > 0 and a finite covering by disjoint balls B 1 , ..., B N (with the sum of radii at most r) Since, by Lemma 7, |v ε | ≥ 1/2 in B(0, 2) \ B(0, 1), D k ≥ d k , and (94) follows from (99) and the estimate U ε ≥ b. Proof of Corollaries 5 and 6. By combining (93) and (94) we obtain the lower bound for F ε in Ω: The conclusions of the above corollaries are obtained by solving the discrete minimization problem of the RHS of (100).
As a direct consequence of Proposition 3 and Lemma 8, we have Corollary 7. There is C > 0 independent of ε s.t. for 1 > ρ > 2δ we have

Existence of the limiting solution
We now return to the proof of Theorems 1 and 2.
Recall that {i ε 1 , ..., i ε d ′ } is a set of distinct elements of {1, ..., M }. We choose ε small enough so that i j 's are independent of ε, thus we may simply denote this set by {i 1 , ..., i d ′ }. In Case I, we have d ′ = d and we may assume that {i 1 , ..., i d ′ } = {1, ..., d}. In Case II, we have d ′ = M . Lemma 7 and Corollary 7 imply that for an appropriate extraction ε = ε n ↓ 0 and for a compact Therefore, when ε → 0, up to a subsequence, there exists We now fix such sequence and a compact K By exactly the same argument as in Proposition 7 we deduce that v ε is bounded in C k (K) for all k ≥ 0 and 1 − |v ε | 2 ≤ C K ε 2 in K.
Consequently, up to subsequence we have for a compact set Now, assume that K is s.t. K ⊂ Ω \ {a i 1 , ..., a i d ′ } but K ∩ ω δ = ∅ (then we are in Case I). Without loss of generality, assume K = B(a k 0 , R), where a k 0 ∈ {a d+1 , ..., a M } and R > 0 is sufficiently small in order to have K ∩ {a 1 , ..., a M } = {a k 0 }.
Assertion 3. of Theorem 2 is is a consequence of Corollary 6.

The macroscopic position of vortices minimizes the Bethuel-Brezis-Hélein renormalized energy
Let us recall briefly the concept of the renormalized energy W g ((b 1 , d 1 ), ..., For small ρ > 0, consider Ω ρ = Ω \ ∪ i B(b i , ρ) and the minimization problem Such problem is studied in detail in [11] (Chapter 1). In particular Bethuel, Brezis and Hélein proved that for small ρ, we have This equality plays an important role in the study done in [11]. In the minimization problem of the classical Ginzburg-Landau functional the vortices (with their degrees) of a minimizer tend to form (up to a subsequence) a minimal configuration for W g . We prove in this section that the (macroscopic) location of the vorticity of minimizers of F ε is related to the minimization problem of W g ((b 1 , ..., b k ), (d 1 , ..., d k )) with b 1 , ..., b k ∈ {a 1 , ..., a M }.
We present here the argument for Case I (Theorem 1). The argument in Case II is analogous. The proof of Assertion 4. relies on two lemmas, providing sharp upper and lower bounds.
Proof. The proof, via construction of a test function, is the same as proof of Lemma VIII.1 in [11].
Thus, combining (108), (109) and using Proposition 2, for ε sufficiently small, we have By Theorem 4 and Corollary 5 we have the following energy expansion: Similarly, applying Theorem 4 to J(ε, ρ) we obtain Here, the local renormalized energyW (α) is given by (90) and is the same in (111) and (112).
which gives the lower bound in the second sub-domain. From (110) and (113) the bound (107) follows.
We next turn to convergence of v ε up to the boundary. It suffices to prove the H 1 -convergence of v ε in Ω ρ = Ω \ ∪ m B(a im , ρ) (for small ρ > 0). We argue by contradiction and we assume that there are some ρ 1 > 0 and η > 0 s.t.
Note that for all ρ ≤ ρ 1 , (114) still holds in Ω ρ . If, in the proof of Lemma 10, we replace (108) by (114) (with ρ 1 replaced by ρ), then we obtain for small ρ a contradiction with Lemma 9. The proof of Theorem 1 is complete. The last assertion of Theorem 2 is obtained along the same lines.

A Proof of Proposition 2
Let x 0 ∈ V R be s.t. B R = B(x 0 , R) ⊂ Ω \ ω δ and assume that x 0 = 0. We follow the proof of Lemma 2 in [10].
here, t will be chosed later.

B Proof of Proposition 3
This appendix is devoted to the proof of Proposition 3. We prove the first assertion: when M ≥ d we have Such a function clearly exists since the compatibility condition deg ∂Ω (g) = d i=1 deg ∂B(a i ,ρ 0 ) (ṽ) is satisfied. Let c 0 = 10 −2 · dist(0, ∂ω). For every 1 ≤ i ≤ M , consider a disc B(a i , c 0 δ) ⊂ ω i δ . By the choice of c 0 , we have dist(∂ω δ , B(a i , c 0 δ)) ≥ c 0 δ. Therefore, using Proposition 2 Consider the test function v ε 0 defined as x − a i ε for x ∈ B(a i , ε) .
(For example, the map considered in Remark I.5 in [11] has these properties).
The necessary test function that satisfies the bound (12) is obtained by rescaling the v ε i 's (in order to have maps defined in balls of size δ) and gluing the rescaled maps withṽ ε 0 .

C Proof of the η-ellipticity Lemma
The main argument in the proof of the η-ellipticity result is the following convexity lemma which is a generalization of Lemma 8 in [9]. The proof of Lemma 11 is given in [15].

Lemma 11. [Convexity Lemma]
Let C be a chord in the closed unit disc, C different from a diameter. Let S be the smallest of two regions enclosed by the chord and the boundary of the disc.