Self-dual partial differential systems and their variational principles:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York, NY
Springer
2009
|
| Series: | Springer monographs in mathematics
|
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 354 S. |
| ISBN: | 9780387848969 |
Staff View
MARC
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| 245 | 1 | 0 | |a Self-dual partial differential systems and their variational principles |c Nassif Ghoussoub |
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Record in the Search Index
| _version_ | 1804138556016295936 |
|---|---|
| adam_text | Contents
Preface
............................................................ ix
1
Introduction
................................................... 1
Parti
CONVEX
ANALYSIS ON
PHASE
SPACE
2
Legendre-Fenchel
Duality on
Phase
Space
........................ 25
2.1
Basic notions
of convex analysis
.............................. 25
2.2
Subdifferentiability of convex functions
........................ 26
2.3
Legendre duality for convex functions
......................... 28
2.4
Legendre transforms of integral functionals
..................... 31
2.5
Legendre transforms on phase space
........................... 32
2.6
Legendre transforms on various path spaces
.................... 38
2.7
Primal and dual problems in convex optimization
................ 45
3
Self-dual Lagrangians on Phase Space
............................ 49
3.1
Invariance
under Legendre transforms up to an automorphism
..... 49
3.2
The class of self-dual Lagrangians
............................ 51
3.3
Self-dual Lagrangians on path spaces
.......................... 55
3.4
Uniform convexity of self-dual Lagrangians
.................... 57
3.5
Regularization of self-dual Lagrangians
........................ 59
3.6
Evolution triples and self-dual Lagrangians
..................... 62
4
Skew-Adjoint Operators and Self-dual Lagrangians
............... 67
4.1
Unbounded skew-symmetric operators and self-dual Lagrangians
.. 67
4.2
Green-Stokes formulas and self-dual boundary Lagrangians
....... 73
4.3
Unitary groups associated to skew-adjoint operators and self-duality
78
5
Self-dual Vector Fields and Their Calculus
........................ 83
5.1
Vector fields derived from self-dual Lagrangians
................ 84
5.2
Examples of
ß-self-dual
vector fields
.......................... 86
„:;
Contents
All
5.3
Operations on self-dual vector fields
........................... 88
5.4
Self-dual vector fields and maximal monotone operators
.......... 91
Part II COMPLETELY SELF-DUAL SYSTEMS AND THEIR
LAGRANGIANS
6
Variational Principles for Completely Self-dual Functionals
......... 99
6.1
The basic variational principle for completely self-dual
functionals
................................................ 99
6.2
Complete self-duality in non-self
adj
oint Dirichlet
problems
.......103
6.3
Complete self-duality and non-potential PDEs in divergence form
.. 107
6.4
Completely self-dual functionals for certain differential systems
... 110
6.5
Complete self-duality and
semilinear
transport equations
.........113
7
Semigroups of Contractions Associated to Self-dual Lagrangians
----119
7.1
Initial-value problems for time-dependent Lagrangians
...........120
7.2
Initial-value parabolic equations with a diffusive term
............125
7.3
Semigroups of contractions associated to self-dual Lagrangians
.... 129
7.4
Variational resolution for gradient flows of semiconvex functions
.. 135
7.5
Parabolic equations with homogeneous state-boundary conditions
.. 137
7.6
Variational resolution for coupled flows and wave-type equations
.. 140
7.7
Variational resolution for parabolic-elliptic variational
inequalities
................................................143
8
Iteration of Self-dual Lagrangians and Multiparameter Evolutions
.. 147
8.1
Self-duality and nonhomogeneous boundary value problems
......148
8.2
Applications to PDEs involving the transport operator
............153
8.3
Initial-value problems driven by a maximal monotone operator
___155
8.4
Lagrangian intersections of convex-concave Hamiltonian systems
.. 161
8.5
Parabolic equations with evolving state-boundary conditions
......162
8.6
Multiparameter evolutions
...................................166
9
Direct Sum of Completely Self-dual Functionals
...................175
9.1
Self-dual systems of equations
................................176
9.2
Lifting self-dual Lagrangians
toAf¡[0,T]
.......................178
9.3
Lagrangian intersections via self-duality
.......................180
10 Semilinear
Evolution Equations with Self-dual Boundary
Conditions
....................................................187
10.1
Self-dual variational principles for parabolic equations
...........187
10.2
Parabolic
semilinear
equations without a diffusive term
...........191
10.3
Parabolic
semilinear
equation with a diffusive term
..............195
10.4
More on skew-adjoint operators in evolution equations
...........199
Contents xiii
Part III SELF-DUAL SYSTEMS AND THEIR ANTISYMMETRIC
HAMILTONIANS
11
The Class of Antisymmetric Hamiltonians
........................205
11.1
The Hamiltonian and co-Hamiltonians of self-dual Lagrangians
.... 206
11.2
Regular maps and antisymmetric Hamiltonians
..................210
11.3
Self-dual functionals
........................................212
12
Variational Principles for Self-dual Functionals and First
Applications
...................................................217
12.1
Ky Fan s
min-max
principle
..................................217
12.2
Variational resolution for general nonlinear equations
............220
12.3
Variational resolution for the stationary Navier-Stokes equations
... 227
12.4
A variational resolution for certain nonlinear systems
............230
12.5
A nonlinear evolution involving a pseudoregular operator
.........232
13
The Role of the Co-Hamiltonian in Self-dual Variational Problems
.. 241
13.1
A self-dual variational principle involving the co-Hamiltonian
.....241
13.2
The Cauchy problem for Hamiltonian flows
....................242
13.3
The Cauchy problem for certain nonconvex gradient flows
........247
14
Direct Sum of Self-dual Functionals and Hamiltonian Systems
......253
14.1
Self-dual systems of equations
................................253
14.2
Periodic orbits of Hamiltonian systems
........................260
14.3
Lagrangian intersections
.....................................267
14.4
Semiconvex Hamiltonian systems
.............................270
15
Superposition of Interacting Self-dual Functionals
.................275
15.1
The superposition in terms of the Hamiltonians
..................275
15.2
The superposition in terms of the co-Hamiltonians
...............278
15.3
The superposition of a Hamiltonian and a co-Hamilonian
.........281
Part IV PERTURBATIONS OF SELF-DUAL SYSTEMS
16
Hamiltonian Systems of Partial Differential Equations
.............287
16.1
Regularity and compactness via self-duality
....................288
16.2
Hamiltonian systems of PDEs with self-dual boundary conditions
.. 289
16.3
Nonpurely diffusive Hamiltonian systems of PDEs
...............299
17
The Self-dual Palais-Smale Condition for Noncoerdve Functionals
.. 305
17.1
A self-dual nonlinear variational principle without coercivity
......306
17.2
Superposition of a regular map with an unbounded linear
operator
..................................................309
17.3
Superposition of a nonlinear map with a skew-adjoint operator
modulo boundary terms
.....................................315
xiv Contents
18 Navier-Stokes
and other Self-dual Nonlinear Evolutions
............319
18.1
Elliptic perturbations of self-dual functionate
...................319
18.2
A self-dual variational principle for nonlinear evolutions
..........323
18.3
Navier-Stokes evolutions
....................................331
18.4 Schrödinger
evolutions
......................................340
18.5
Noncoercive nonlinear evolutions
.............................342
References
.........................................................345
Index
.............................................................353
|
| any_adam_object | 1 |
| author | Ghoussoub, Nassif 1953- |
| author_GND | (DE-588)137025246 |
| author_facet | Ghoussoub, Nassif 1953- |
| author_role | aut |
| author_sort | Ghoussoub, Nassif 1953- |
| author_variant | n g ng |
| building | Verbundindex |
| bvnumber | BV035269085 |
| classification_rvk | SK 540 |
| classification_tum | MAT 350f |
| ctrlnum | (OCoLC)316247688 (DE-599)DNB990588009 |
| dewey-full | 515.353 |
| dewey-hundreds | 500 - Natural sciences and mathematics |
| dewey-ones | 515 - Analysis |
| dewey-raw | 515.353 |
| dewey-search | 515.353 |
| dewey-sort | 3515.353 |
| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| format | Book |
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| id | DE-604.BV035269085 |
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| indexdate | 2024-07-09T21:30:04Z |
| institution | BVB |
| isbn | 9780387848969 |
| language | English |
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| spelling | Ghoussoub, Nassif 1953- Verfasser (DE-588)137025246 aut Self-dual partial differential systems and their variational principles Nassif Ghoussoub New York, NY Springer 2009 XIV, 354 S. txt rdacontent n rdamedia nc rdacarrier Springer monographs in mathematics Elliptische Differentialgleichung (DE-588)4014485-9 gnd rswk-swf Variationsrechnung (DE-588)4062355-5 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 gnd rswk-swf Partielle Differentialgleichung (DE-588)4044779-0 s Variationsrechnung (DE-588)4062355-5 s DE-604 Elliptische Differentialgleichung (DE-588)4014485-9 s 1\p DE-604 Erscheint auch als Online-Ausgabe 978-0-387-84897-6 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017074486&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis 1\p cgwrk 20201028 DE-101 https://d-nb.info/provenance/plan#cgwrk |
| spellingShingle | Ghoussoub, Nassif 1953- Self-dual partial differential systems and their variational principles Elliptische Differentialgleichung (DE-588)4014485-9 gnd Variationsrechnung (DE-588)4062355-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| subject_GND | (DE-588)4014485-9 (DE-588)4062355-5 (DE-588)4044779-0 |
| title | Self-dual partial differential systems and their variational principles |
| title_auth | Self-dual partial differential systems and their variational principles |
| title_exact_search | Self-dual partial differential systems and their variational principles |
| title_full | Self-dual partial differential systems and their variational principles Nassif Ghoussoub |
| title_fullStr | Self-dual partial differential systems and their variational principles Nassif Ghoussoub |
| title_full_unstemmed | Self-dual partial differential systems and their variational principles Nassif Ghoussoub |
| title_short | Self-dual partial differential systems and their variational principles |
| title_sort | self dual partial differential systems and their variational principles |
| topic | Elliptische Differentialgleichung (DE-588)4014485-9 gnd Variationsrechnung (DE-588)4062355-5 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd |
| topic_facet | Elliptische Differentialgleichung Variationsrechnung Partielle Differentialgleichung |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=017074486&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| work_keys_str_mv | AT ghoussoubnassif selfdualpartialdifferentialsystemsandtheirvariationalprinciples |