Vortices in the magnetic Ginzburg-Landau Model:
Saved in:
| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhauser
2007
|
| Series: | Progress in nonlinear Differential equations and their applications
70 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XII, 322 S. graph. Darst. |
| ISBN: | 0817643168 9780817643164 |
Staff View
MARC
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| 245 | 1 | 0 | |a Vortices in the magnetic Ginzburg-Landau Model |c Etienne Sandier ; Sylvia Serfaty |
| 264 | 1 | |a Boston [u.a.] |b Birkhauser |c 2007 | |
| 300 | |a XII, 322 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
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| 338 | |b nc |2 rdacarrier | ||
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Record in the Search Index
| _version_ | 1804138626794127360 |
|---|---|
| adam_text | Contents
Preface
χι
1
Introduction
1
1.1
The Model
.......................... 2
1.1.1
Vortices
........................ 3
1.1.2
Critical Fields
..................... 4
1.2
Questions Addressed in this Book
.............. 5
1.3
Ginzburg-Landau with and without
Magnetic Field: A Comparison
............... 6
1.4
Plan of the Book
....................... 7
1.4.1
Essential Tools
.................... 8
1.4.2
Minimization Results
................. 10
1.4.3
Branches of Local Minimizers
............ 17
1.4.4
Results on Critical Points
.............. 21
2
Physical Presentation of the Model
—
Critical Fields
25
2.1
The Ginzburg-Landau Model
................ 25
2.1.1
Nondimensionalizing
................. 26
2.1.2
Dimension Reduction
................ 27
2.1.3
Gauge
Invariance
................... 28
2.2
Notation
............................ 29
2.3
Constant States in R2
.................... 29
,2.4
Periodic Solutions
...................... 30
2.5
Vortex Solutions
....................... 31
2.5.1
Approximate Vortex
................. 31
2.5.2
The Energy of the Approximate Vortex
...... 33
2.5.3
The Critical Line HC1
................ 35
2.6
Phase Diagram
........................ 36
2.6.1
Bounded Domains
.................. 37
vj Contents
3
First Properties of Solutions to the Ginzburg-Landau
Equations
39
3.1
Minimizing the Ginzburg-Landau Energy
......... 39
3.1.1
Coulomb Gauge
................... 40
3.1.2
Restriction to
Ω
................... 41
3.1.3
Minimization of GL
................. 42
3.2
Euler-Lagrange Equations
.................. 43
3.3
Properties of Critical Points
................. 46
3.4
Solutions in the Plane
.................... 50
3.4.1
Degree Theory
.................... 50
3.4.2
The Radial Degree-One Solution
.......... 52
3.4.3
Solutions of Higher Degree
............. 53
3.5
Blow-up Limits
........................ 54
4
The Vortex-Balls Construction
59
4.1
Main Result
.......................... 60
4.2
Ball Growth
.......................... 61
4.3
Lower Bounds for S1-valued Maps
............. 65
4.4
Reduction to S1-valued Maps
................ 71
4.4.1
Radius of a Compact Set
.............. 71
4.4.2
Lower Bound on Initial Balls
............ 72
4.4.3
Proof of Theorem
4.1 ................ 73
4.5
Proof of Proposition
4.7................... 76
4.5.1
Initial Set
....................... 76
4.5.2
Construction of the Appropriate Initial Collection
78
5
Coupling the Ball Construction to the Pohozaev Identity
and Applications
83
5.1
The Case of
Ginzburg-Landau
without Magnetic Field
. . 83
5.2
The Case of Ginzburg-Landau with Magnetic Field
.... 96
5.3
Applications
.......................... 102
6
Jacobian Estimate
117
6.1
Preliminaries
......................... 118
6.2
Proof of Theorem
6.1..................... 120
6.3
A Corollary
.......................... 123
Contents
vii
7
The Obstacle Problem
127
7.1
F-Convergence
........................ 128
7.2
Description of
μ*
....................... 130
7.3
Upper Bound
......................... 134
7.3.1
The Space Hl and the Green Potential
...... 135
7.3.2
The Energy-Splitting Lemma
............ 136
7.3.3
Configurations with Prescribed Vortices
...... 137
7.3.4
Choice of the Vortex Configuration
......... 142
7.4
Proof of Theorems
7.1
and
7.2 ............... 150
7.4.1
Proof of Theorem
7.1,
Item
1) ........... 150
7.4.2
Proof of Theorem
7.2 ................ 152
8
Higher Values of the Applied Field
155
8.1
Upper Bound
.........................157
8.2
Lower Bound
.........................160
9
The Intermediate Regime
165
9.1
Main Result
..........................165
9.1.1
Motivation
......................166
9.1.2
Γ-
Convergence in the Intermediate Regime
.... 168
9.2
Upper Bound: Proof of Proposition
9.1...........172
9.3
Proof of Theorem
9.1.....................175
9.3.1
Energy-Splitting Lower Bound
...........177
9.3.2
Lower Bound on the Annulus
............181
9.3.3
Compactness and Lower Bounds Results
......187
9.3.4
Completing the Proof of Theorem
9.1 .......200
9.4
Minimization with Respect to
η
...............201
10
The Case of a Bounded Number of Vortices
207
10.1
Upper Bound
.........................207
10.2
Lower Bound
.........................213
11
Branches of Solutions
219
11.1
The Renormalized Energy wn
................ 219
11.2
Branches of Solutions
.................... 224
11.3
The Local Minimization Procedure
............. 226
11.4
The Case
N = 0....................... 227
11.5
Upper Bound for
inîUNGs
.................. 228
11.6
Minimizing Sequences Stay Away from dU^
........ 230
Ущ
Contents
11.7
inf^
Ge
is Achieved
.....................235
11.8
Proof of Theorem
11.1....................236
12
Back to Global Minimization
243
12.1
Global
Minimizers
Close to HC1
...............243
12.2
Possible Generalization: The Case where
Λ
is not Reduced
to a Point
...........................248
13
Asymptotics for Solutions
253
13.1
Results and Examples
.................... 255
13.1.1
The Divergence-Free Condition
........... 256
13.1.2
Result in the Case with Magnetic Field
...... 259
13.1.3
The Case without Magnetic Field
.......... 265
13.2
Preliminary Results
..................... 269
13.3
Proof of Theorem
13.1,
Criticality Conditions
....... 275
13.4
Proof of Theorem
13.1,
Regularity Issues
.......... 278
13.5
The Case without Magnetic Field
.............. 280
14
A Guide to the Literature
283
14.1
Ginzburg-Landau without Magnetic Field
.........283
14.1.1
Static Dimension
2
Case in a Simply Connected
Domain
........................283
14.1.2
Vortex Solutions in the Plane
............285
14.1.3
Other Boundary Conditions
.............285
14.1.4
Weighted Versions
..................286
14.1.5
Construction of Solutions
..............286
14.1.6
Fine Behavior of the Solutions
...........286
14.1.7
Stability of the Solutions
..............287
14.1.8
Jacobian Estimates
..................287
14.1.9
Dynamics
.......................287
14.2
Higher Dimensions
......................288
14.2.1
T-Convergence Approach
..............288
14.2.2
Minimizers and Critical Points Approach
.....289
14.2.3
Inverse Problems
...................289
14.2.4
Dynamics
.......................290
14.3
Ginzburg-Landau with Magnetic Field
...........290
14.3.1
Dependence on
к
...................290
14.3.2
Vortex Solutions in the Plane
............291
14.3.3
Static Two-Dimensional Model
...........292
Contents ix
14.3.4 Dimension
Reduction ................
295
14.3.5 Models
with Pinning Terms
............. 295
14.3.6
Higher Dimensions
.................. 295
14.3.7
Dynamics
....................... 296
14.3.8
Mean-Field Models
.................. 296
14.4
Ginzburg-Landau in Nonsimply Connected Domains
. . . 296
15
Open Problems
299
Index
321
|
| any_adam_object | 1 |
| author | Sandier, Etienne Serfaty, Sylvia |
| author_GND | (DE-588)133106594 (DE-588)133106616 |
| author_facet | Sandier, Etienne Serfaty, Sylvia |
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| author_sort | Sandier, Etienne |
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| classification_rvk | SK 560 |
| ctrlnum | (OCoLC)82148950 (DE-599)BVBBV035319093 |
| dewey-full | 537.6/23 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 537 - Electricity and electronics |
| dewey-raw | 537.6/23 |
| dewey-search | 537.6/23 |
| dewey-sort | 3537.6 223 |
| dewey-tens | 530 - Physics |
| discipline | Physik Mathematik |
| format | Book |
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| id | DE-604.BV035319093 |
| illustrated | Illustrated |
| indexdate | 2024-07-09T21:31:11Z |
| institution | BVB |
| isbn | 0817643168 9780817643164 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-017123699 |
| oclc_num | 82148950 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-19 DE-BY-UBM |
| owner_facet | DE-355 DE-BY-UBR DE-19 DE-BY-UBM |
| physical | XII, 322 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Birkhauser |
| record_format | marc |
| series | Progress in nonlinear Differential equations and their applications |
| series2 | Progress in nonlinear Differential equations and their applications |
| spelling | Sandier, Etienne Verfasser (DE-588)133106594 aut Vortices in the magnetic Ginzburg-Landau Model Etienne Sandier ; Sylvia Serfaty Boston [u.a.] Birkhauser 2007 XII, 322 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in nonlinear Differential equations and their applications 70 Mathematik Superconductivity Mathematics Flussschlauch (DE-588)4326682-4 gnd rswk-swf Magnetfeld (DE-588)4074450-4 gnd rswk-swf Ginzburg-Landau-Theorie (DE-588)4157357-2 gnd rswk-swf Ginzburg-Landau-Gleichung (DE-588)4157356-0 gnd rswk-swf Ginzburg-Landau-Gleichung (DE-588)4157356-0 s Ginzburg-Landau-Theorie (DE-588)4157357-2 s Flussschlauch (DE-588)4326682-4 s Magnetfeld (DE-588)4074450-4 s DE-604 Serfaty, Sylvia Verfasser (DE-588)133106616 aut Progress in nonlinear Differential equations and their applications 70 (DE-604)BV007934389 70 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017123699&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Sandier, Etienne Serfaty, Sylvia Vortices in the magnetic Ginzburg-Landau Model Progress in nonlinear Differential equations and their applications Mathematik Superconductivity Mathematics Flussschlauch (DE-588)4326682-4 gnd Magnetfeld (DE-588)4074450-4 gnd Ginzburg-Landau-Theorie (DE-588)4157357-2 gnd Ginzburg-Landau-Gleichung (DE-588)4157356-0 gnd |
| subject_GND | (DE-588)4326682-4 (DE-588)4074450-4 (DE-588)4157357-2 (DE-588)4157356-0 |
| title | Vortices in the magnetic Ginzburg-Landau Model |
| title_auth | Vortices in the magnetic Ginzburg-Landau Model |
| title_exact_search | Vortices in the magnetic Ginzburg-Landau Model |
| title_full | Vortices in the magnetic Ginzburg-Landau Model Etienne Sandier ; Sylvia Serfaty |
| title_fullStr | Vortices in the magnetic Ginzburg-Landau Model Etienne Sandier ; Sylvia Serfaty |
| title_full_unstemmed | Vortices in the magnetic Ginzburg-Landau Model Etienne Sandier ; Sylvia Serfaty |
| title_short | Vortices in the magnetic Ginzburg-Landau Model |
| title_sort | vortices in the magnetic ginzburg landau model |
| topic | Mathematik Superconductivity Mathematics Flussschlauch (DE-588)4326682-4 gnd Magnetfeld (DE-588)4074450-4 gnd Ginzburg-Landau-Theorie (DE-588)4157357-2 gnd Ginzburg-Landau-Gleichung (DE-588)4157356-0 gnd |
| topic_facet | Mathematik Superconductivity Mathematics Flussschlauch Magnetfeld Ginzburg-Landau-Theorie Ginzburg-Landau-Gleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017123699&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV007934389 |
| work_keys_str_mv | AT sandieretienne vorticesinthemagneticginzburglandaumodel AT serfatysylvia vorticesinthemagneticginzburglandaumodel |