Vortices in the magnetic ginzburg-landau model

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Title:

Personal Author:

Sandier, Etienne

Series:

Progress in nonlinear differential equations and their applications 70

Publication Information:

New York, NY Birkh�auser Boston, 2007

Physical Description:

xii, 322 p. : ill. ; 24 cm.

ISBN:

9780817643164

Subject Term:

Superconductivity -- Mathematics

Added Author:

Serfaty, Sylvia

Available:*

Library	Item Barcode	Call Number	Material Type	Item Category 1	Status
Searching... PSZ JB	30000010222241	QC611.92 S26 2007	Open Access Book	Book	Searching... Unknown

More than ten years have passed since the book of F. Bethuel, H. Brezis and F. H´ elein, which contributed largely to turning Ginzburg-Landau equations from a renowned physics model into a large PDE research ?eld, with an ever-increasing number of papers and research directions (the number of published mathematics papers on the subject is certainly in the several hundreds, and that of physics papers in the thousands). Having ourselves written a series of rather long and intricately - terdependent papers, and having taught several graduate courses and mini-courses on the subject, we felt the need for a more uni?ed and self-contained presentation. The opportunity came at the timely moment when Ha¨ ?m Brezis s- gested we should write this book. We would like to express our gratitude towards him for this suggestion and for encouraging us all along the way. As our writing progressed, we felt the need to simplify some proofs, improvesomeresults,aswellaspursuequestionsthatarosenaturallybut that we had not previously addressed. We hope that we have achieved a little bit of the original goal: to give a uni?ed presentation of our work with a mixture of both old and new results, and provide a source of reference for researchers and students in the ?eld.

Preface	p. xi
1 Introduction	p. 1
1.1 The Model	p. 2
1.1.1 Vortices	p. 3
1.1.2 Critical Fields	p. 4
1.2 Questions Addressed in this Book	p. 5
1.3 Ginzburg-Landau with and without Magnetic Field: A Comparison	p. 6
1.4 Plan of the Book	p. 7
1.4.1 Essential Tools	p. 8
1.4.2 Minimization Results	p. 10
1.4.3 Branches of Local Minimizers	p. 17
1.4.4 Results on Critical Points	p. 21
2 Physical Presentation of the Model - Critical Fields	p. 25
2.1 The Ginzburg-Landau Model	p. 25
2.1.1 Nondimensionalizing	p. 26
2.1.2 Dimension Reduction	p. 27
2.1.3 Gauge Invariance	p. 28
2.2 Notation	p. 29
2.3 Constant States in R[superscript 2]	p. 29
2.4 Periodic Solutions	p. 30
2.5 Vortex Solutions	p. 31
2.5.1 Approximate Vortex	p. 31
2.5.2 The Energy of the Approximate Vortex	p. 33
2.5.3 The Critical Line H[subscript c1]	p. 35
2.6 Phase Diagram	p. 36
2.6.1 Bounded Domains	p. 37
3 First Properties of Solutions to the Ginzburg-Landau Equations	p. 39
3.1 Minimizing the Ginzburg-Landau Energy	p. 39
3.1.1 Coulomb Gauge	p. 40
3.1.2 Restriction to [Omega]	p. 41
3.1.3 Minimization of GL	p. 42
3.2 Euler-Lagrange Equations	p. 43
3.3 Properties of Critical Points	p. 46
3.4 Solutions in the Plane	p. 50
3.4.1 Degree Theory	p. 50
3.4.2 The Radial Degree-One Solution	p. 52
3.4.3 Solutions of Higher Degree	p. 53
3.5 Blow-up Limits	p. 54
4 The Vortex-Balls Construction	p. 59
4.1 Main Result	p. 60
4.2 Ball Growth	p. 61
4.3 Lower Bounds for S[superscript 1]-valued Maps	p. 65
4.4 Reduction to S[superscript 1]-valued Maps	p. 71
4.4.1 Radius of a Compact Set	p. 71
4.4.2 Lower Bound on Initial Balls	p. 72
4.4.3 Proof of Theorem 4.1	p. 73
4.5 Proof of Proposition 4.7	p. 76
4.5.1 Initial Set	p. 76
4.5.2 Construction of the Appropriate Initial Collection	p. 78
5 Coupling the Ball Construction to the Pohozaev Identity and Applications	p. 83
5.1 The Case of Ginzburg-Landau without Magnetic Field	p. 83
5.2 The Case of Ginzburg-Landau with Magnetic Field	p. 96
5.3 Applications	p. 102
6 Jacobian Estimate	p. 117
6.1 Preliminaries	p. 118
6.2 Proof of Theorem 6.1	p. 120
6.3 A Corollary	p. 123
7 The Obstacle Problem	p. 127
7.1 [Gamma]-Convergence	p. 128
7.2 Description of [micro]	p. 130
7.3 Upper Bound	p. 134
7.3.1 The Space H[subscript 1] and the Green Potential	p. 135
7.3.2 The Energy-Splitting Lemma	p. 136
7.3.3 Configurations with Prescribed Vortices	p. 137
7.3.4 Choice of the Vortex Configuration	p. 142
7.4 Proof of Theorems 7.1 and 7.2	p. 150
7.4.1 Proof of Theorem 7.1, Item 1)	p. 150
7.4.2 Proof of Theorem 7.2	p. 152
8 Higher Values of the Applied Field	p. 155
8.1 Upper Bound	p. 157
8.2 Lower Bound	p. 160
9 The Intermediate Regime	p. 165
9.1 Main Result	p. 165
9.1.1 Motivation	p. 166
9.1.2 [Gamma]-Convergence in the Intermediate Regime	p. 168
9.2 Upper Bound: Proof of Proposition 9.1	p. 172
9.3 Proof of Theorem 9.1	p. 175
9.3.1 Energy-Splitting Lower Bound	p. 177
9.3.2 Lower Bound on the Annulus	p. 181
9.3.3 Compactness and Lower Bounds Results	p. 187
9.3.4 Completing the Proof of Theorem 9.1	p. 200
9.4 Minimization with Respect to 72	p. 201
10 The Case of a Bounded Number of Vortices	p. 207
10.1 Upper Bound	p. 207
10.2 Lower Bound	p. 213
11 Branches of Solutions	p. 219
11.1 The Renormalized Energy w[subscript n]	p. 219
11.2 Branches of Solutions	p. 224
11.3 The Local Minimization Procedure	p. 226
11.4 The Case N = 0	p. 227
11.5 Upper Bound for inf[subscript Un] - G[subscript epsilon]	p. 228
11.6 Minimizing Sequences Stay Away from [Characters not reproducible]U[subscript N]	p. 230
11.7 inf[subscript UN] G[subscript epsilon] is Achieved	p. 235
11.8 Proof of Theorem 11.1	p. 236
12 Back to Global Minimization	p. 243
12.1 Global Minimizers Close to H[subscript c1]	p. 243
12.2 Possible Generalization: The Case where A is not Reduced to a Point	p. 248
13 Asymptotics for Solutions	p. 253
13.1 Results and Examples	p. 255
13.1.1 The Divergence-Free Condition	p. 256
13.1.2 Result in the Case with Magnetic Field	p. 259
13.1.3 The Case without Magnetic Field	p. 265
13.2 Preliminary Results	p. 269
13.3 Proof of Theorem 13.1, Criticality Conditions	p. 275
13.4 Proof of Theorem 13.1, Regularity Issues	p. 278
13.5 The Case without Magnetic Field	p. 280
14 A Guide to the Literature	p. 283
14.1 Ginzburg-Landau without Magnetic Field	p. 283
14.1.1 Static Dimension 2 Case in a Simply Connected Domain	p. 283
14.1.2 Vortex Solutions in the Plane	p. 285
14.1.3 Other Boundary Conditions	p. 285
14.1.4 Weighted Versions	p. 286
14.1.5 Construction of Solutions	p. 286
14.1.6 Fine Behavior of the Solutions	p. 286
14.1.7 Stability of the Solutions	p. 287
14.1.8 Jacobian Estimates	p. 287
14.1.9 Dynamics	p. 287
14.2 Higher Dimensions	p. 288
14.2.1 [Gamma]-Convergence Approach	p. 288
14.2.2 Minimizers and Critical Points Approach	p. 289
14.2.3 Inverse Problems	p. 289
14.2.4 Dynamics	p. 290
14.3 Ginzburg-Landau with Magnetic Field	p. 290
14.3.1 Dependence on [kappa]	p. 290
14.3.2 Vortex Solutions in the Plane	p. 291
14.3.3 Static Two-Dimensional Model	p. 292
14.3.4 Dimension Reduction	p. 295
14.3.5 Models with Pinning Terms	p. 295
14.3.6 Higher Dimensions	p. 295
14.3.7 Dynamics	p. 296
14.3.8 Mean-Field Models	p. 296
14.4 Ginzburg-Landau in Nonsimply Connected Domains	p. 296
15 Open Problems	p. 299
Index	p. 321

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