The Pennsylvania State University The Graduate School ASYMPTOTIC ANALYSIS OF GINZBURG-LANDAU SUPERCONDUCTIVITY MODEL

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1 The Pennsylvania State University The Graduate School ASYMPTOTIC ANALYSIS OF GINZBURG-LANDAU SUPERCONDUCTIVITY MODEL A Dissertation in Mathematics by Oleksandr Misiats c 01 Oleksandr Misiats Submitted in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy December 01

2 The dissertation of Oleksandr Misiats was reviewed and approved by the following: Leonid Berlyand Professor of Mathematics Dissertation Advisor, Committee Chair Alberto Bressan Eberly Chair Professor of Mathematics Anna L. Mazzucato Associate Professor of Mathematics Qiang Du Verne M. Willaman Professor of Mathematics Sarah Shandera Assistant Professor of Physics Anatole Katok Raymond N. Shibley Professor of Mathematics Head of the Department of Mathematics Signatures are on file in the Graduate School.

3 Abstract The focus of this dissertation is the asymptotic analysis of variational problems and PDEs, related to Ginzburg-Landau theory of superconductivity. In Chapter 1 we describe the physical phenomenon of superconductivity, and focus on the mathematical aspects of modeling this phenomenon via variational problems. Special attention is paid to the choice of boundary conditions, which enables us to model vortices, an inherent feature of superconductivity. While these variational problems are motivated by physics and contribute to a deeper understanding of certain physical notions, they are interesting from a mathematical standpoint and lead to new mathematical challenges. A number of novel mathematical concepts and methods was developed in the process of their rigorous analysis, which may be useful in the analysis of other mathematical problems. In Chapter (with L.Berlyand and V.Rybalko, [1] and []) we consider the minimization of magnetic Ginzburg-Landau functional in both simply connected and doubly connected domains. The order parameter is subject to so-called semistiff boundary conditions: unit modulus and prescribed topological degrees on the domain boundary. We also study the vorticity structure of Ginzburg-Landau minimizers. In the case of a simply connected domain, we obtain that the minimizers with vortices (zeros) exist if and only if the parameter λ (called the coupling constant) is lower or equal than the critical value λ cr = 1. We also observe that the vortices of these minimizers are strictly inside the domain for λ close to 1. On the contrary, in doubly connected domains, we observe a different picture. Namely, we establish the existence of minimizers with nearboundary vortices, which converge to the boundary of the domain as λ approaches λ cr. Additionally, we describe the limiting positions of the vortices on the boundary via the solutions of an auxiliary scalar linear boundary value problem. Chapter 3 (the work [3]) studies the minimization problem for the simplified Ginzburg-Landau functional in doubly connected domain. This minimization probiii

4 lem is a subject to semi-stiff boundary conditions with prescribed degrees p and q on the outer and inner boundaries respectively. It is known that if p q, there are no global minimizers for simplified Ginzburg-Landau functional with such boundary conditions. Therefore, we additionally prescribe the degree in the bulk (approximate bulk degree), introduced in [4], to be d. The work [4] established the sufficient conditions on the existence of local Ginzburg-Landau minimizers with semi-stiff boundary conditions, which are given in terms of p, q and d. Chapter 3 complements the result of [4] by providing the necessary conditions for the existence of such minimizers. Chapters 4 and 5 (the works [5] and [6]) are devoted to modeling composite superconductors. This work was motivated by the physical models of vortex pinning (i.e., fixing the positions of vortices), which is done by introducing inclusions into a homogeneous superconductor. Mathematically, composite superconductors are modeled via Ginzburg-Landau type functional with a piecewise constant pinning term a in the potential (a u ), which takes two different values in the medium and in the inclusions. In Chapter 4 (in collaboration with P. Mironescu and M. Dos Santos) we study the minimization problem for such functional subject to Dirichlet boundary conditions with zero degree on the boundary. We obtain the homogenized description of Ginzburg-Landau minimizers in the limit of large number of inclusions and small ε (where ε is the inverse Ginzburg-Landau parameter). The subsequent chapter 5 (with M. Dos Santos) is focused on modeling a superconductor with finitely many small superconducting inclusions in the presence of vortices. We show that even the inclusions of vanishingly small size (e.g. shrinking to single points) capture the vortices of minimizers. This way we reduce the problem of finding the locations of the vortices to a discrete minimization problem for a finite-dimensional functional of renormalized energy. iv

5 Table of Contents List of Figures Acknowledgments ix x Chapter 1 Introduction Phenomenon of superconductivity Discovery of superconductivity Meissner effect. Emergence of vortices Ginzburg-Landau energy functional Modeling homogeneous superconductors Simplified GL functional with Dirichlet Boundary Conditions From Dirichlet to Semi-Stiff Boundary Conditions Magnetic Ginzburg-Landau model with Semi-Stiff Boundary Conditions Necessary conditions for the existence of local Ginzburg-Landau minimizers with semi-stiff BC Ginzburg-Landau model for composite superconductors I Ginzburg-Landau model with semi-stiff boundary conditions 17 Chapter Minimization of magnetic Ginzburg-Landau functional 18.1 Minimization problem in a simply connected domain: bulk vortices Introduction v

6 .1. Regularity of critical points Existence/nonexistence of the minimizers Asymptotic behavior of vortices Numerical experiments Minimization problem in a doubly connected domain: near boundary vortices Introduction Preliminaries Upper bound construction Existence/nonexistence of minimizers Lower bound Proof of the key lemma Near boundary vortices and δ-like behavior of currents Explicit formula for energy bounds Chapter 3 Necessary conditions for the existence of local Ginzburg-Landau minimizers with prescribed degrees Introduction and main results Properties of abdeg Proof of Theorem : Energy decomposition Model problem Model problem in thin domains Model problem in general case Proof of Theorem II Modeling composite superconductors 98 Chapter 4 Homogenization results for Ginzburg Landau Functional with Discontinuous Pinning Term Introduction General weighted Ginzburg-Landau type functionals Uniform convergence of v ε to A corollary of Theorem More on the convergence of v ε The Ginzburg-Landau functional with a periodic pinning term The dilute limit λ vi

7 4.3. The case λ = 1, δ = ε The case λ = 1, ε δ The case λ = 1, δ ε Chapter 5 Vortices in Ginzburg-Landau model with small pinning domains Introduction and main results Main tools Properties of U ε Upper Bounds Identifying bad discs A model problem: one inclusion Uniform convergence of ˆv ε to 1 away from inclusions Distribution of Energy in B(0, ρ ) δ Convergence in C (K) for a compact K s.t. K ω = The bad discs Convergence in Hloc 1 (R \ {α 1,..., α κ }) Information about the limit v Uniqueness of zeros Summary Renormalized energy for the model problem Minimization among S 1 -valued maps away from the inclusion Energy estimates for S 1 -valued maps around the inclusion Upper bound for the energy Lower bound The function g 0 and the points {α 1,..., α d0 } minimize the renormalized energy Proofs of Theorems 5.1 and Locating bad inclusions Existence of the limiting solution The macroscopic position of vortices minimizes the Bethuel- Brezis-Helein renormalized energy Chapter 6 Conclusions and Prospectives 184 Appendix Adimensionalizing Ginzburg-Landau energy Proof of Proposition Proof of Proposition vii

8 4 Proof of the η-ellipticity Lemma Bibliography 195 viii

9 List of Figures 1.1 Experimental image of a vortex lattice Location of vortices in the unit disk, d = 1,, Splitting of vortices, f 1 (z) = 1(z ), d = z+ 3 ix

10 Acknowledgments First and foremost, I would like to thank my advisor, Professor Leonid Berlyand, for introducing me to the study of Ginzburg-Landau theory and for exposing me to a large variety of cutting edge problems in modern mathematics. I am indebted to him for his guidance and genuine interest in my professional development. His unwillingness to accept anything but my best efforts have made the current work possible. I would also like to thank Dr. Volodymyr Rybalko for the fruitful collaboration and for always having his door open to answer my questions, technical and otherwise. I am grateful to Professors Petru Mironescu and Felix Otto for the collaboration and hospitality during my visits to the University of Lyon 1 and University of Bonn, respectively. Special thanks goes to Professor Soren Bartels for the help with the numerical computations in my thesis. I would also like to thank Dr. Konstantin Lipnikov for his guidance during my internship at Los Alamos National Laboratory. I would like to thank the members of my committee, Professor Alberto Bressan, Professor Anna Mazzucato, Professor Qiang Du and Professor Sarah Shandera, who have graciously taken time from their busy schedules to carefully review this work. I also express my gratitude to my undergraduate advisor, Professor Oleksandr Stanzhitskii, who s guidance in conjunction with high standards and relentless commitment laid the foundation of my future career as a mathematician. Finally, I greatly appreciate the endless love of my mother, who continues to inspire and support all of my endeavors, who started me off right with a passion for learning and a passion for perfection. x

11 Chapter 1 Introduction 1.1 Phenomenon of superconductivity Discovery of superconductivity In 1911, while studying the electrical properties of metals at very low temperatures, the Dutch physicist Kamerlingh Onnes was measuring the resistance of a pure mercury wire while gradually lowering the temperature. As expected, the resistance was slowly decreasing with the decrease in temperature. Surprisingly, once the temperature reached 4. K, the resistance suddenly dropped to an absolute zero and remained zero as he continued lowering the temperature. According to Onnes, Mercury has passed into a new state, which on account of its extraordinary electrical properties may be called the Superconductive state [7]. Most metals become superconductors at sufficiently low temperatures, including tin, aluminium and various metallic alloys. In contrast, noble metals, such as gold, and ferromagnetic metals do not possess this property. Some of the physical properties of superconductors, such as the critical temperature etc., may vary from material to material. However, there is a class of properties of superconductors that are independent of the given material and thus indicate that superconductivity is a thermodynamic phase. One of the key distinguishing properties of superconductors is Meissner effect, described in the following subsection.

12 1.1. Meissner effect. Emergence of vortices One of the most important characteristic properties of all superconductors is the repulsion of weak magnetic fields from the bulk of a superconductor. This property was discovered by the German physicists W. Meissner and R. Ochsenfeld in 1933 and is called Meissner effect. In order to describe Meissner effect, introduce a complex-valued function u Hloc 1 (R3 ), such that n s = e u describes the local density of superconducting particles (called Cooper pairs), e - electron charge. This wave function of superconducting particles u is called the order parameter. Assume Ω R 3 is a superconducting sample at sub-critical temperature. Let A tot Hloc 1 (R3, R 3 ) be a vector potential of the total magnetic field B = curla tot. Under the assumption that the sample Ω is purely superconducting, i.e. u const, we may obtain using Maxwell equations that B = 1 δ B (1.1) called London equation. It follows from London equation that there cannot be any uniform nonzero magnetic field inside the superconductor. Moreover, any external field decays exponentially when moving inside the superconductor. If the surface of the superconductor is flat, the magnetic field depends only on one coordinate (perpendicular to the surface), that is B = B(x). Then, from (1.1) we have B(x) = B(0)e x δ (1.) B(0) - initial field on the surface. Therefore the depth of penetration of the magnetic field is of order δ, which is called the London penetration depth [8, 9, 10, 11]. Depending on the behavior of superconductors in sufficiently strong external magnetic fields, superconductors fall in two main categories. Superconductor is called to be of type I if there exists H c, the critical value of the external magnetic field, such that the sample is a pure superconductor for H < H c (and thus the Meissner effect takes place) while for H > H c the sample is at normal state. In other words, type I superconductors do not have vortices and have a first-order (discontinuous) phase transition from the normal state to the superconducting

13 3 state. Unlike type I superconductors, type II superconductors exhibit second-order (continuous) phase transitions with the formation of vortices and vortex lattices. Type II superconductors have more than one critical field. If the external magnetic field is below the first critical value H c1, the superconductor completely repels the external field. Once the external magnetic field reaches H c1, the first vortices appear. In superconductors, vortices are the localized defects, at which the superconducting state is destroyed, and the magnetic field penetrates the bulk of superconductor. The order parameter has a nontrivial winding (nonzero winding number) around this defect. Between H c1 and the second critical field H c, the superconducting and normal phases (in the form of vortices) coexist in the sample. This is called the mixed state, or incomplete Meissner effect. The number of vortices increases with the increase of the external magnetic field. Due to vortex-vortex and vortex-boundary interactions, vortices arrange themselves in patterns, called Abrikosov lattice. Figure 1.1. Experimental image of a vortex lattice Once the external magnetic field reaches H c, vortices start to merge with each other. As the result, the superconductivity is destroyed the in the bulk

14 4 of a superconductor, while it still may be present close to the boundary, called boundary superconductivity. Finally, once the external magnetic field reaches H c3, the superconductivity is destroyed completely, and the sample is at normal state. Vortices determine the key physical properties of superconductors. As mentioned in [1], the behavior of vortices promises to permanently alter the way we think about superconductivity and its potential applications. One of such applications is the superconducting magnetic shield, used, for example, in medical imaging to eliminate the external magnetic signals and the magnetic field of the Earth, in order to measure the weak magnetic signals emitted by the human brain [13]. When constructing this shield, it is important to determine at which points on the shield s surface the sufficiently strong magnetic field is most likely to penetrate inside through the vortex lines, as well as to determine the value of this field. Another practical issue in application of superconductors is to minimize the motion of vortices caused by the electric field. Such motion dissipates the energy and destroys the zero-resistivity state [14], [15]. To this end, various models of pinning (fixing the positions) of vortices are used, e.g. introducing impurities into a homogeneous superconductor. Inspired by these practical applications, in this thesis we perform the variational analysis of Ginzburg-Landau energy functional, which is the key mathematical tool in superconductivity theory. Namely, we establish the existence of Ginzburg-Landau minimizers with vortices, determine the locations of the vortices depending on the geometry of the domain, and develop a model of vortex pinning. While motivated by physics, this analysis leads to a number of mathematical challenges, e.g. locating singularities of solutions for the problems with the loss of compactness, which we address in this thesis. Note that the class of problems with the loss of compactness plays an important role in various areas of mathematics ([16] and references therein), such as the Yamabe problem in differential geometry Ginzburg-Landau energy functional In 1950s, the physicists Lev Landau and Vitaliy Ginzburg suggested the following variational model to describe phase transitions in superconductors. According to their theory, the order parameter u, describing the state of a superconductor at

15 5 sub-critical temperature, and the induced magnetic field potential A must minimize (in the steady state) the following energy functional [ E = E 0 + Ω m ( u ie c Au ) + a u + b u 4 ] dx + 1 curla H ext. 8π R 3 (1.3) Here E 0 is the energy of the sample at normal (non-superconducting) state, H ext is the external magnetic field, m is electron mass, e is electron charge, is Plank constant, a and b are material parameters (which also depend on the temperature). The terms a u + b u 4 in the expression (1.3) are essentially the first two terms in the asymptotic expansion of the energy potential in the powers of u. The term u ie c Au describes the spatial variations of u, and curla H ext is the total magnetic field. Adimentionalizing (1.3) using linear change of variables (Appendix 1), we get where GL[u, A] = 1 Ω u iau + 1 E = Ẽ0 + c 0 GL[u, A] R 3 curla H + 1 4ε Ω (1 u ) (1.4) Here, we keep the same notation (u, A) for the rescaled quantities. The material parameter ε is the ratio of two important physical quantities, the coherence length ξ and penetration depth δ. The penetration depth δ (given by (1.)) is the measure of penetration of the external magnetic field in the bulk of a superconductor, while the coherence length is the measure of penetration of the superconducting wave function into non-superconducting regions. The inverse of ε is called Ginzburg- Landau parameter. In this dissertation we will be working with a two-dimensional reduction of the functional (1.4): F ε [u, A] := 1 Ω u iau + 1 R curla H ext + 1 4ε Ω (1 u ) (1.5) Here Ω R may be viewed as a cross section of a cylindrical superconducting

16 6 wire. Such wires have parallel vortex tubes with nearly D structure, as suggested by the experimental pictures ([17], Fig I.4). 1. Modeling homogeneous superconductors 1..1 Simplified GL functional with Dirichlet Boundary Conditions Rigorous mathematical results on Ginzburg-Landau functional (1.5) started to appear predominantly in the late 1980s- early 1990s (e.g. [18], [19], [0]). However, the pioneering work in developing a mathematical theory for D Ginzburg-Landau functional was the work of Bethuel, Brezis and Helein [1]. authors suggested to consider a simplified Ginzburg-Landau functional E ε [u] := 1 obtained from (1.5) by setting A H 0. Ω In this work, the u + 1 (1 u ) (1.6) 4ε Ω This model is more amenable to rigorous analysis yet it captures the essential physical features of (1.5). Besides, the model (1.6) has a number of applications in modeling superfluids and liquid crystals. Clearly, the functional (1.6) has trivial constant minimizers α S 1. Thus, in order to get nontrivial minimizers with vortices, i.e. to model the effect of the external magnetic field H ext, the authors suggested to impose the Dirichlet boundary condition u = g on the boundary of a smooth simply connected domain Ω. The function g is smooth and satisfies g = 1 on Ω, modeling the purely superconducting state on the domain boundary, and has a non-zero Brouwer degree (winding number) deg(g, Ω) := 1 π Ω g g dσ = d. (1.7) τ In (1.7) and in what follows a b = i (a b bā) and (a, b) = 1 (a b + bā) stand for the wedge and scalar products of complex numbers a and b. We shall use also notations and for the classical Cauchy operators = 1 ) z z z ( x 1 + i x and = ( 1 ) z x 1 i x (where z = x1 + ix ), and ϕ = ( ϕ x, ϕ x 1 ). As noted in the book of Bethuel, Brezis and Helein, [1],... it is striking to

17 7 see that the degree d = deg(g, Ω) of the boundary condition creates the same quantized vortices as a magnetic field in type-ii superconductors or as angular rotation in superfluids. One of the main results of [1] may be summarized in the following theorem: Theorem 1.1. Let u ε be a minimizer of (1.6) in Hg 1 := {u H 1 (Ω), u = g on Ω} with g satisfying g = 1 and (1.7) (such minimizer clearly exists). Then there exist d distinct points a 1,..., a d Ω and a function u C (Ω \ {a 1,..., a d }, S 1 ) s.t. u ε u in C (Ω \ {a 1,..., a d }) as ε 0; u solves u = u u, called the harmonic map equation; The following energy expansion holds as ε 0: E ε [u ε ] = πd ln ε + W g (a 1,..., a d ) + o 1 (ε) (1.8) Here W g (a 1,..., a d ) is a finite dimensional functional of renormalized energy, which gives the locations of the vortices a 1,..., a d. These vortices are away from each other and from Ω. 1.. From Dirichlet to Semi-Stiff Boundary Conditions The work of Bethuel, Brezis and Helein stimulated the further development of mathematical theory of Ginzburg-Landau type problems with Dirichlet boundary conditions. However, it was mentioned in [1] that in physical situations the Dirichlet condition is not realistic. The primary reason is that u is not a measurable physical quantity, unlike u, which is the density of Cooper pairs. Thus, the natural question that one may ask is: is it possible to relax Dirichlet BC by considering m ε := inf g inf E ε [u]? (1.9) u Hg 1(Ω) Here, the first infimum is taken over the set of all g H 1/ ( Ω) satisfying g = 1 a.e. on Ω and deg(g, Ω) = d (as noted in [0], degree is a well-defined integer

18 8 for g H 1/ ( Ω, S 1 )). The quantity m ε can be equivalently written as m ε = inf u J d E ε [u], (1.10) where J d := {u H 1 (Ω; C), u = 1 on Ω, deg(u, Ω) = d}. (1.11) The minimization problem (1.10) was first considered in [], [3] and [4]. Minimization of (1.6) in class (1.11) leads to the Ginzburg-Landau equation u = 1 ε u(1 u ) (1.1) with so called semi-stiff boundary conditions (BC) u = 1 and u u ν = 0 on Ω. (1.13) These boundary conditions are intermediate between Dirichlet and Neumann in the following sense: any solution u H 1 (Ω; C) of (1.1)-(1.13) is sufficiently regular [4], so it can be written as u = u e iψ (locally) near the boundary. Then (1.13) means that Dirichlet boundary conditions are prescribed for the modulus, u = 1 on Ω, and Neumann conditions are prescribed for the phase, ψ = 0 on Ω [4]. ν Prescribing u = 1 on the boundary enables us to model the purely superconducting boundary layer. The natural question, arising in this modeling, is the following: is the minimization problem for (1.6) in (1.11) well posed, i.e. is the infimum in (1.10) achieved? It was observed in ([4]) that the infimum in (1.10) may not be attained since the class J d is not weakly H 1 closed. The primary reason is that the topological degree, given by (1.7), may change in weak H 1 limits. The heuristic explanation of this fact is as follows: if u n u in H 1 (G), then u n u in H 1/ ( G) while u n u in H 1/ ( G). Thus, (1.7) is essentially a product (in the sense of H 1/ H 1/ duality) of two weakly convergent sequences, which may not converge. Besides, it is not difficult to construct a minimizing sequence in J d, whose degree changes in the limit. For example, if Ω is a unit disc and d = 1, we may construct a sequence of Mobius conformal maps a n (z) = z an 1 za n with a n = 1 1. While for every n 1 a n n(z) is in J 1, the weak H 1 limit of a n (z) is a

19 9 constant 1, which is not in J 1. Moreover, it follows from the pointwise identity u = Jacu + 4 u and the Jacobian degree formula Jacu = πdeg(u, G), z G which hold for any u J 1, that a n is a minimizing sequence for (1.6) in J 1 [4], therefore, for any fixed ε > 0 the infimum of (1.6) in J 1 is not attained. Thus, the existence of Ginzburg-Landau minimizers in (1.11), as well as the existence of stable nontrivial (nonconstant) solutions of (1.1) with semi-stiff boundary conditions, is a nontrivial issue. This issue will be addressed in Chapters 1 and of this dissertation in both simply and doubly connected domains. In doubly connected domains G = Ω \ ω, which model superconductors with holes, we may prescribe the degrees on both inner and outer boundaries. This leads to considering the minimization problem for (1.6) in J p,q := {u H 1 (G; C), u = 1 on G, deg(u, Ω) = p, deg(u, ω) = q}. (1.14) for some integers p and q. If p q, it is not difficult to see ([4]) that there are no global minimizers of (1.6) in J p,q. The case p = q = 1 was studied in [3], [4] and [5]. The existence of vortexless minimizers was established in sufficiently thin domains. It was also shown in [5] that for sufficiently small ε there are no global minimizers of (1.6) in J 11 in thick domains. The subsequent work [4] established the existence of stable solutions of (1.1) with nearboundary vortices, or local minimizers in (1.14). The work [6] generalized this result for the case of the domains with several connected components Magnetic Ginzburg-Landau model with Semi-Stiff Boundary Conditions In Chapter 1 (with L. Berlyand and V. Rybalko) we focus on the search for global Ginzburg-Landau minimizers in (1.11). We consider the minimization problem for the following functional G λ [u, A] = 1 u iau dx + 1 curla dx + λ G R 8 G ( u 1) dx, (1.15) (which is obtained from (1.5) by setting H ext 0). Here G C is a smooth bounded domain, and λ :=. One of the main characteristics of the functional ε

20 10 (1.15) is its invariance under the gauge transformations (u, A) (e iϕ u, A + ϕ), ϕ H (R ). This property allows one to reduce the study of (1.15) to the functional with R replaced by Ω (see, e.g., [7], Chapter 3), where Ω is a simply connected domain containing G: F λ [u, A] = 1 u iau dx + 1 G Ω curla dx + λ ( u 1) dx. (1.16) 8 If G is simply connected, we can choose Ω = G in (1.16). Making use of the gauge invariance of F λ [u, A] one more time, we can assume that (u, A) is in the Coulomb gauge, i.e. diva = 0 in Ω (1.17) A ν = 0 on Ω. In Section.1 we study the existence/nonexistence of minimizers of (1.16) in the class G J d := {(u, A) H 1 (Ω; C) H 1 (Ω; R ); u = 1 a.e. on Ω, A satisfies (1.17)}, (1.18) where Ω is a simply connected domain. The necessary and sufficient conditions for the existence of minimizers of (1.16) in (1.18) are given in the following Theorem 1.. Let d be a given nonzero integer. Then (a) For 0 < λ 1, the minimizer of (1.16) in (1.18) is attained; (b) For λ > 1, the minimizer of (1.16) in (1.18) is never attained. Theorem 1. shows that, unlike simplified Ginzburg-Landau functional, the magnetic Ginzburg-Landau functional (1.16) has nontrivial minimizers with semistiff boundary conditions. Theorem 1. is proved by induction in d. The main difficulty in its proof is to obtain sharp energy estimates for the functions in J d through the energies of the functions from J d+1 or J d 1, which is done via a proper choice of test functions in these classes. An important case in the analysis of the minimization problem (1.16)-(1.18) is the case λ = 1, called the critical case. In this case the minimizers of (1.16)-(1.18) can be found through Taubes factorization procedure ([0] and references therein) and are called the Bogomolniy solutions. We show that the minimizers for λ < 1

21 11 (whose existence was established in Theorem 1.) up to a subsequence converge to Bogomolniy solutions as λ 1, and the limiting locations of their vortices may be determined via minimization of a finite-dimensional functional. In particular, these vortices are located strictly inside G. Remark 1.1. From the physical point of view, the value λ = 1 (ε = 1 ) for the functional (1.3) in an unbounded domain Ω = R 3 is considered to be critical for the following reason: for λ < 1 the material is a Type I superconductor, while for λ > 1 the material is a Type II superconductor. However, we cannot make the same conclusion for the functional (1.16), since this functional is minimized in a bounded domain Ω with the boundary condition u = 1 on Ω. This condition models a purely superconducting layer on the boundary, which means that the superconductor effectively behaves like Type II superconductor (i.e. has a vortex state) even for λ < 1. Section. focuses on the magnetic Ginzburg-Landau energy functional in a doubly connected domain G = Ω \ ω: F λ [u, A] := 1 Ω curla + 1 The minimization of (1.19) is done in the class G ( ia)u + λ 8 G (1 u ) (1.19) J 01 := {(u, A) H 1 (G, C) H 1 (Ω, R ), u = 1 on G, deg(u, Ω) = 1, deg(u, ω) = 0.} (1.0) The principal difference of this problem from the minimization in a simply connected domain is the presence of the extra term ω curla 0. This term leads to nonexistence of minimizers in the critical case λ = 1: Theorem 1.3. (a) For 0 < λ < 1, the minimizer of (1.19) in (1.0) is always attained; (b) For λ 1, the minimizer of (1.19) in (1.0) is never attained. It is worth mentioning that the minimization problems for F λ with semi-stiff boundary conditions fall into the class of variational problems with possible lack of compactness for minimizing sequences. In particular, this class contains semilinear problems with critical nonlinearities. These problems are closely related to

22 1 the Yamabe problem in differential geometry (see [8, 9, 16, 30] and references therein). For this problem, it is known that if the infimum of a certain functional is lower than some magic level, this infimum is attained, and thus the compactness is restored. This approach is widely used for various problems with lack of compactness [16]. In our case, we show that m λ := inf (u,a) J01 F λ [u, A] is attained if and only if m λ < π. We then proceed with establishing that m λ < π for 0 < λ < 1 and m λ π (in fact, m λ = π) for λ 1. The key result of the Section. is the description of the behavior of minimizers as λ 1 : Theorem 1.4. Let (u λ, A λ ), 0 < λ < 1 be the minimizer of (1.19)- (1.0). Then (i) u λ has exactly one vortex ξ λ and ξ λ ξ Ω as λ 1 ; (ii) The location of ξ Ω is determined by minimization problem inf, ξ Ω, where v : G R is the unique solution of v = v in G; v = 1 on ω; v = 0 on Ω. v(ξ) ν (1.1) (iii) j λ τ = (iu λ, u λ ia λ u λ ) τ πδ ξ currents). in D ( Ω) (δ-like behavior of boundary Theorem 1.4 shows that if the values of λ are close to 1 (from below) Ginzburg- Landau minimizers may have vortices near the boundary (or nearboundary vortices). Theorem 1.4 is proved via obtaining tight upper and lower bounds for the energy of minimizers Necessary conditions for the existence of local Ginzburg-Landau minimizers with semi-stiff BC Let G be a doubly connected domain, i.e. G = Ω \ ω, and p, q Z are integers. Consider the minimization problem for (1.6) in J pq := {u H 1 (G), u = 1 on G, deg(u, Ω) = p, deg(u, ω) = q.} (1.)

23 13 The work [4] established the existence of local minimizers of (1.6) in (1.). The main tool for establishing the local minimizers was the notion of approximate bulk degree, introduced in [4]: abdeg(u) := 1 u ( x1 V x u x V x1 u) dx. (1.3) π G where the scalar function V solves V = 0 in G with V = 1 on Ω and V = 0 on ω. The key property of abdeg(u) is that, unlike the degree, it is preserved in weak- H 1 limits. We may now introduce the following open subset of J pq : J (d) pq = {u J pq ; d 1/ < abdeg(u) < d + 1/}. (1.4) The main result of [4] was that for any integers p, q and d > 0 (d < 0) with d max{p, q} (d min{p, q}) the infimum of (1.6) in (1.4) is attained, and thus it is a local minimizer of (1.6) in (1.). The results in Chapter 3 complement the work [4] by providing the necessary conditions for the existence of nontrivial (nonconstant) minimizers: d > 0, d > min{p, q} if p q, and d > 0, d p if p = q (or d < 0, d < max{p, q} if p q, and d < 0, d p if p = q). In other words, we prove the following theorem Theorem 1.5. Let d > 0, d min{p, q} (or d < 0, d max{p, q}) and either p d or q d. Let u be a weak limit of a minimizing sequence for the problem min u J (d) pq Then u J (d) pq E ε[u], with E ε given by (1.6) (such minimizing sequence always exists). when ε is sufficiently small. 1.3 Ginzburg-Landau model for composite superconductors We consider a simplified Ginzburg-Landau functional E a ε [u] := 1 Ω u + 1 ( u a ) dx (1.5) 4ε Ω where the piecewise constant function a is called the pinning term. This function is different from 1 on disjoint subdomains ω δ Ω, called the pinning domains or

24 14 inclusions. We study the energy minimization for (1.5) with Dirichlet boundary data g such that g = 1 on Ω. The functionals of this type arise in models of superconductivity for composite superconductors. The domain Ω can be viewed as a cross-section of a multifilamentary wire with a number of thin superconducting filaments, modeled by the pinning domains. Such multifilamentary wires are widely used in industry, including magnetic energy-storing devices, transformers and power generators [31], [3]. 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 ([33], [34]). 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 (1.5) 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. The functional (1.5) with non-constant a(x) was proposed by Rubinstein in [35] as a model of pinning vortices for Ginzburg-Landau minimizers. Shortly after, Andre and Shafrir [36] 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 [37]. 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 1 if x Ω \ ω a(x) =, b if x ω here ω is a simply connected open set s.t. ω Ω. The main objective of [37] 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 [38]. An ε-dependent but

25 15 continuous pinning term a ε (x) was studied by Aftalion, Sandier and Serfaty in [39]. The work [40] studies the case of a smooth a with finite number of isolated zeros, and in [41] 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., [33], [4]. Chapter 4 (with P.Mironescu and M. Dos Santos) deals with the rapidly oscillating discontinuous pinning term a δ (x), which takes values 0 < b < 1 on a two-dimensional δ-periodic array of inclusions inside Ω, and value 1 otherwise. The geometric parameter δ is assumed to depend on the material parameter ε and goes to zero as ε 0. We additionally assume the Dirichlet boundary data u = g on Ω satisfies the condition deg(g, Ω) = 0. Our approach is based on Lassoued and Mironescu s idea [37] of factorizing the minimizer u ε as u ε = U ε v ε, where U ε is a real-valued minimizer of (1.5) with the boundary condition U ε 1 on Ω, and v ε is a minimizer of the weighted Ginzburg-Landau type functional F ε [v] := 1 with α ε W 1, (Ω, [b, 1]) and β ε L (Ω, [b, 1]). Ω α ε v + 1 β 4ε ε (1 v ) (1.6) Ω Theorem 1.6. Let v ε be a minimizer of (1.6) in the class with Dirichlet boundary data v ε = g on Ω with deg(g, Ω) = 0 and satisfying the minimal regularity requirements g H 1 ( Ω). Then v ε satisfies v ε 1 both in L ( Ω) and H 1 (Ω). In particular, this theorem implies that for sufficiently small ε the minimizers v ε are vortexless, hence we may write v ε = v ε e iϕε for some real single-valued phase ϕ ε, which solves a linear equation. This enables us to apply the linear homogenization theory, and, depending on the asymptotic relation between δ and ε (i.e. δ >> ε, δ = ε and δ << ε) we obtain different homogenization limits for v ε as ε 0. For example, in the most physically relevant case δ >> ε we have Theorem 1.7. Assume the pinning term a = a δ is periodic with period δ >> ε. Let u ε be a minimizer of (1.5) in the class with Dirichlet boundary data u ε = g on Ω with deg(g, Ω) = 0. Then ρ ε = u ε â in L (Ω), where â is the average of a over the periodicity cell, and ϕ ε ϕ in H 1 (Ω), where ϕ solves the homogenized problem div(a ϕ ) = 0 in Ω. Here A is the homogenized matrix for a ( x δ ) IdR.

26 16 In the subsequent Chapter 5 (with M. Dos Santos) we consider the minimization problem for (1.5) in the presence of vortices. We take the pinning term a to be different from 1 only in finitely many disjoint pinning domains of size δ, and impose the nonzero degree boundary condition deg(g, Ω) = d > 0. Our main ln δ(ε) 3 result is that under the assumption 0 the minimizers u ln ε ε of (1.5) in the limit of small ε have exactly d vortices, which are located inside the pinning inclusions. In a particular case when the number of inclusions M is larger than d (number of vortices) there are exactly d distinct inclusions with vortices, containing one vortex each. The surprising fact is that the asymptotic (as ε 0) location of the vortex inside the pinning inclusion depends only on its geometric shape but not on the macroscopic location of the inclusion and, more importantly, on the boundary condition g. We also show that the configuration of the inclusions with vortices minimizes the renormalized energy of Bethuel, Brezis and Helein among the discrete set of all d-element subsets of the set of M inclusions. Thus, the problem of finding the locations of the vortices of minimizers is reduced to a discrete minimization problem for finite-dimensional functional of renormalized energy. The distribution of vortices in the case M < d is also described in this chapter.

27 Part I Ginzburg-Landau model with semi-stiff boundary conditions

28 Chapter Minimization of magnetic Ginzburg-Landau functional.1 Minimization problem in a simply connected domain: bulk vortices.1.1 Introduction Let Ω R be a bounded simply connected domain with a smooth boundary. We consider the Ginzburg-Landau energy functional F λ [u, A] = 1 u iau dx + 1 Ω Ω curla dx + λ ( u 1) dx. (.1) 8 where u H 1 (Ω; C) is the order parameter, A H 1 (R ; R ) is the vector potential of the induced magnetic field, and λ > 0 is the coupling constant ( λ/ is the Ginzburg-Landau parameter). Using the invariance of F λ [u, A] under the gauge transformations (u, A) (e iϕ u, A + ϕ), ϕ H (Ω), and choosing an appropriate ϕ, we can assume that (u, A) is in the Coulomb gauge, i.e. diva = 0 in Ω (.) A ν = 0 on Ω. Ω

29 19 Note that the induced magnetic field h = curla = A / x 1 A 1 / x, the current j = (iu, u iau) (.3) and u are gauge invariant. Besides, the topological degree of u on Ω, which is (well-)defined for any u H 1/ ( Ω; S 1 ) (see, e.g., [0]) by the classical formula (1.7) is also gauge invariant. We study the critical points of F λ [u, A] in J = {(u, A) H 1 (Ω; C) H 1 (Ω; R ); u = 1 a.e. on Ω, A satisfies (.35)}. (.4) They are the solutions of the system of Euler-Lagrange equations ( ia) u + λ u( u 1) = 0 h = j in Ω, (.5) subject to the boundary conditions u = 1, A ν = 0, h = 0 and j ν = 0 on Ω. (.6) The boundary condition u = 1 on Ω assumes the presence of superconducting current on the boundary. As shown in [0], the space J is disconnected, its connected components J d, d Z, are classified by the topological degree of the order parameter u on the boundary Ω, J = d Z J d, J d = {(u, A) J ; deg(u, Ω) = d}. The minimization problem for (.1) in J d (d 0) with the additional Dirichlet condition on the tangential component of the current was considered in [43]. Prescribing the tangential component of the current allows one to control the tangential derivative of the order parameter u. This, in turn, yields that the topological degree of the order parameter is preserved when passing to the weak limit in a minimizing sequence and thus shows that the minimization problem is well-posed. By

30 0 contrast, minimizing over the entire space J d results in the homogeneous boundary condition for the normal component of the current, j ν = 0, which means that the current is tangential to the boundary. This boundary condition, however, leads to a problem with a possible lack of compactness, that is minimizers in J d for d 0 may not exist. The principal difficulty is due to the fact that J d is not closed with respect to the weak H 1 -convergence (see [4], [], [44]). Prior to the present paper, the existence of minimizers in J d has only been established in [0] for the integrable (self-dual) case λ = 1. A comprehensive study of existence / nonexistence of minimizers in bounded domains with prescribed degrees on the connected components of the boundary was performed in [4], [4], [], [3], [5] for the simplified GL functional E λ [u] = F λ [u, 0]. The critical points of the energy functional (.9) in the entire space, when Ω = R, were studied in [45], [46], [19], [47], [48]. In this section we establish the existence/nonexistence of the solutions of the minimization problem m d (λ) := inf{f λ [u, A]; (u, A) J d }. (.7) The principal result is Theorem.1. (i) The infimum in (.7) is always attained for 0 < λ < 1, that is any minimizing sequence converges (up to subsequence) to a global minimizer of (.7). (ii) If λ > 1 then the infimum m d (λ) is never attained unless d = 0 (in the latter case we have trivial minimizers u Const S 1, A 0). Note that for the critical value λ = 1 of the coupling constant minimizers exist and form a d -parametric family which is parameterized by (arbitrary) locations of the vortices in Ω (see Proposition. below). We also study the the asymptotic behavior of minimizers as λ 1 0. Namely, we investigate the vorticity structure of the minimizers and determine the limiting locations of vortices. We show, in particular, that minimizers converge, up to a subsequence, to a minimizer (u, A) of (.7) for λ = 1 that satisfies curla dx max (see Theorem.3 below). Ω

31 1.1. Regularity of critical points As proved in [0], every (u, A) J solving (.5) satisfies u C (Ω; C) and A C (Ω; R ). We show that solutions of (.5) with boundary conditions (.39) (critical points of F λ [u, A] in J ) have the C -regularity up to the boundary. Proposition.1. Let (u, A) J be a solution of (.5) - (.39). Assume that the boundary Ω is C -smooth. Then u C ( Ω; C) and A C ( Ω; R ). Proof. Split u into the sum u = v+w, v solving v = ia u+ A u+ λ u( u 1) in Ω with the boundary condition v = 0 on Ω and w solving w = 0 in Ω subject to the boundary condition w = u on Ω. By the Sobolev embedding v L q (Ω) for every 1 < q <, therefore v C α ( Ω) for every 0 < α < 1. On the other hand, since u H 1/ ( Ω; S 1 ), and therefore u VMO( Ω; S 1 ), we have w(x) 1 as x Ω (see [49]). Thus u is continuous on Ω and u 1, (.8) the inequality u 1 in Ω is obtained by applying the maximum principle to the equation 1 u = u iau + λ u ( u 1) in Ω, (.9) which is a consequence of the first equation in (.5). Now, the second equation in (.5) implies that h = curla H 1 (Ω). Moreover, from (.5) we have div( u h) + h = 0 in the subdomain of Ω where u > 0. (.10) Let ξ Ω. Consider a smooth cut-off function θ such that θ(x) = 1 for x < 1 and θ(x) = 0 for x >, and set θ ε (x) = θ((x ξ)/ε). Due to (.10) and (.39), the function θ ε h satisfies (θ ε h) (θ ε h) = div T ε + u θ ε h in Ω (.11) θ ε h = 0 on Ω, (we assume ε > 0 is sufficiently small so that u 1/ on Ω supp θ ε ), where T ε = (1 u ) (θ ε h) + u h θ ε. By the standard elliptic estimates,

32 θ ε h W 1,p C p ( T ε L p + u θ ε L h H 1) for every p > 1. Then, choosing ε small enough so that u 1 1/(C p ) on Ω supp θ ε, we see that θ ε h W 1,p C p u θ ε L ( h L p + h H 1). Thus hθ ε W 1,p (Ω) and therefore h W 1,p (Ω B ε (ξ)), where B ε (ξ) = {x; x ξ < ε}. Since ε > 0 can be chosen independent of ξ, we have h W 1,p (Ω) for every p > 1. It follows from (.9) and the second equation in (.5) that u = u 3 h + λ u ( u 1) in the subdomain of Ω where u > 0. Also we have u = 1 on Ω. This implies that u W,p (Ω) for every p > 1, since we know that h W 1,p (Ω) and u C (Ω; C) H 1 (Ω) L (Ω; C). Similarly, rewriting equation (.10) as h = u h + u 1 u h and taking into account the facts that h = 0 on Ω and h C (Ω) W 1,p (Ω) for every p > 1, we conclude that h W,p (Ω) for every p > 1. Then, by bootstrapping one shows u, h C k ( Ω) for every k = 1,,..., therefore, in view of (.35) and the second equation in (.5), A C ( Ω; R ), u C ( Ω; C)..1.3 Existence/nonexistence of the minimizers Following a remarkable observation of Bogomol nyi [50] (see also [0]), F λ [u, A] can be written as where and F λ [u, A] = πdeg(u, Ω) + F + [u, A] 1 λ 8 = πdeg(u, Ω) + F [u, A] 1 λ 8 F + [u, A] = u Ω z + A ia 1 u dx + 1 F [u, A] = u Ω z A + ia 1 u dx + 1 Ω Ω Ω Ω ( u 1) dx (.1) ( u 1) dx, (.13) u 1 curla + dx, (.14) curla u 1 dx. (.15)

33 3 The functionals F λ [u, A] and F ± [u, A] are lower semicontinuous with respect to the weak convergence in H 1 (Ω; C) H 1 (Ω; R ). By the Sobolev embedding, the last term in (.41) and (.13) is continuous with respect to the weak-h 1 convergence; finally, deg(u, Ω) is integer valued and continuous in J supplied with the strong topology of H 1 (Ω; C) H 1 (Ω; R ) but it is neither continuous nor lower (upper) semicontinuous with respect to the weak-h 1 convergence in J (inherited from H 1 (Ω; C) H 1 (Ω; R )). Therefore, the attainability of infimum in (.7) is a nontrivial problem. As mentioned above, the complete description of the minimizers F λ [u, A] in J d in the integrable (self-dual) case λ = 1 was established in [0]. In this case minimizers form a d -parametric family. This is quite a drastic contrast with the case of simplified Ginzburg-Landau functional, where there are no minimizers for every λ > 0 unless d = 0. When λ = 1, (.41)-(.13) imply that m d (1) π d. Assuming for the sake of definiteness d > 0, the equality m d (1) = πd yields the system of the first order PDEs u + 1(A z ia 1 )u = 0 in Ω, (.16) curla + 1 ( u 1) = 0 By Taubes procedure ([19]) of factorizing u into the product of the holomorphic part a(z; ξ 1 )... a(z; ξ d ) and the factor e ϕ/, (.16) leads to the single second-order PDE for ϕ, ϕ + a(z; ξ 1 )... a(z; ξ d ) e ϕ = 1 in Ω (.17) ϕ = 0 on Ω, where a(z; ξ) is a conformal map from Ω onto B 1 = {x C; x < 1}, such that a(ξ; ξ) = 0, i.e. a(z, ξ) can be written as a(z; ξ) = f(z) f(ξ) 1 f(ξ)f(z) (.18) for some fixed conformal map f from Ω onto B 1. Notice that in the critical case λ = 1 the points ξ 1,..., ξ d, that are the vortices of the minimizer, can be chosen arbitrarily in Ω.

34 4 Proposition.. ([0]) Assume d > 0, then m d (1) = πd and (i) for every ξ 1,..., ξ d Ω (u, A) = (a(z; ξ 1 )... a(z; ξ d )e ϕ/, 1 ϕ) with ϕ solving (.45) is a minimizer of (.7) for λ = 1; (ii) any minimizer of (.7) for λ = 1 can be represented as (u, A) = (γ a(z; ξ 1 )... a(z; ξ d ) e ϕ/, 1 ϕ) for some γ = const S 1, ξ 1,..., ξ d Ω, and ϕ solving (.45). The proof of the attainability of the infimum in (.7) for λ < 1 uses the weak lower semicontinuity of F λ [u, A], F ± [u, A] and the continuity of the last term in (.41)-(.13). It is based on a comparison argument whose key ingredient is the following Lemma. Lemma.1. Let (u, A) J d such that u 1 in Ω. Then (i) there exists (v, B) J d+1 such that v 1 in Ω, F + [v, B] F + [u, A] and ( v 1) dx > ( u 1) dx; (.19) Ω Ω (ii) there exists (v, B) J d 1 such that v 1 in Ω, F [v, B] F [u, A] and (.19) holds. Proof. Let ξ be a Lebesgue point of u such that u(ξ) > 0. Let a(x; ξ) be a conformal map from Ω onto the unit disk given by (.48) (in particular, a(ξ; ξ) = 0). Consider the map v = ua(x; ξ)e φ/ (with real valued φ H (Ω)) and the vector field B = A + B (with B H 1 (Ω; R ) such that v z + B ib 1 v = a(x; ξ)e φ/( u z + A ia 1 u ) in Ω, (.0) curlb + 1 ( v 1) = curla + 1 ( u 1) in Ω (.1) φ = 0 on Ω. Clearly, if B is in the Coulomb gauge then (v, B) Jd+1. Note that (.0) is satisfied when B = 1 φ (which is in the Coulomb gauge) and with this choice