Nonlinear oscillations of Hamiltonian PDEs:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Boston [u.a.]
Birkhäuser
2007
|
| Series: | Progress in non-linear differential equations and their applications
74 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Physical Description: | XIV, 180 S. graph. Darst. |
| ISBN: | 9780817646806 |
Staff View
MARC
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| 100 | 1 | |a Berti, Massimiliano |e Verfasser |0 (DE-588)133631656 |4 aut | |
| 245 | 1 | 0 | |a Nonlinear oscillations of Hamiltonian PDEs |c Massimiliano Berti |
| 264 | 1 | |a Boston [u.a.] |b Birkhäuser |c 2007 | |
| 300 | |a XIV, 180 S. |b graph. Darst. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
| 338 | |b nc |2 rdacarrier | ||
| 490 | 1 | |a Progress in non-linear differential equations and their applications |v 74 | |
| 650 | 4 | |a Opérateur hamiltonien | |
| 650 | 4 | |a Oscillations non linéaires | |
| 650 | 4 | |a Systèmes hamiltoniens | |
| 650 | 4 | |a Équations aux dérivées partielles | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 4 | |a Hamiltonian operator | |
| 650 | 4 | |a Hamiltonian systems | |
| 650 | 4 | |a Nonlinear oscillations | |
| 650 | 0 | 7 | |a Hamiltonsches System |0 (DE-588)4139943-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Periodische Lösung |0 (DE-588)4199269-6 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Nichtlineare Wellengleichung |0 (DE-588)4042104-1 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Verzweigung |g Mathematik |0 (DE-588)4078889-1 |2 gnd |9 rswk-swf |
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| 999 | |a oai:aleph.bib-bvb.de:BVB01-016553218 | ||
Record in the Search Index
| _version_ | 1804137737019719680 |
|---|---|
| adam_text | Contents
1
Finite
Dimension............................................... 1
1.1 The Lyapunov Center Theorem............................... 4
1.2 The Weinstein-Moser and Fadell-Rabinowitz
Resonant
Center
Theorems................................................. 7
1.2.1 The Variational Lyapunov-Schmidt
Reduction
............ 13
1.2.2
Solution of the Range Equation
........................ 14
1.2.3
Solution of the Bifurcation Equation
.................... 15
1.2.4
Proof of the
Weinstein-Moser
Theorem
................. 16
1.2.5
Proof of the Fadell-Rabinowitz Theorem
................ 20
2
Infinite Dimension
............................................. 29
2.1
The Lyapunov Center Theorem for PDEs
....................... 31
2.2
Completely Resonant Wave Equations
......................... 33
2.3
The Case
ρ
Odd
........................................... 35
2.3.1
The Variational Lyapunov-Schmidt Reduction
............ 36
2.3.2
The Range Equation
.................................. 37
2.3.3
The Bifurcation Equation
............................. 41
2.3.4
The Mountain Pass Argument
.......................... 43
2.4
The Case
ρ
Even
........................................... 48
2.5
Multiplicity
............................................... 53
2.6
The Small-Divisor Problem
.................................. 54
3
A Tutorial in Nash-Moser Theory
................................ 59
3.1
Introduction
............................................... 59
3.2
An Analytic Nash-Moser Theorem
............................ 60
3.3
A Differentiable Nash-Moser Theorem
........................ 66
4
Application to the Nonlinear Wave Equation
...................... 73
4.1
The Zeroth-Order Bifurcation Equation
........................ 75
4.2
The Finite-Dimensional Reduction
............................ 77
4.2.1
Solution of the (£>2)-Equation
......................... 77
4.3
Solution of the Range Equation
............................... 80
xiv Contents
4.3.1 The Nash-Moser
Scheme
............................. 83
4.4
Solution of the (^l)-Equation
................................ 94
4.5
The Linearized Operator
..................................... 98
4.5.1
Decomposition of
C„................................. 98
4.5.2
Step
1 :
Inversion of
D
................................ 100
4.5.3
Step
2:
Inversion of Cn
............................... 104
5
Forced Vibrations
..............................................
Ill
5.1
The Forcing Frequency
ω
є
Q
...............................
Ill
5.2
The Variational Lyapunov-Schmidt Reduction
.................. 113
5.2.1
The Range Equation
.................................. 115
5.2.2
The Bifurcation Equation
............................. 117
5.3
Monotone
ƒ............................................... 119
5.3.1
Step
1:
the L°° Estimate
.............................. 119
5.3.2
Step
2:
the
Я1
Estimate
............................... 122
5.4
Nonmonotone
ƒ........................................... 124
5.4.1
Step
1 :
the L2k Estimate
.............................. 129
5.4.2
Step
2:
the L°° Estimate
............................... 130
5.4.3
Step
3:
The
#
Estimate
.............................. 132
5.4.4
The Maximum Principle
............................ 133
Appendix A Hamiltonian PDEs
.................................... 139
A.
1
The Nonlinear
Schrödinger
Equation
.......................... 139
A.2 The Beam Equation
......................................... 139
A.3 The KdV Equation
......................................... 140
A.4 The
Euler
Equations of Hydrodynamics
........................ 140
Appendix
В
Critical Point Theory
.................................. 145
B.
1
Preliminaries
.............................................. 145
B.2 Minima
................................................... 146
B.3 The Minimax Idea
.......................................... 147
B.4 The Mountain Pass Theorem
................................. 149
Appendix
С
Free Vibrations of Nonlinear Wave Equations: A Global
Result
............................................... 155
Appendix
Đ
Approximation of Irrationals by Rationals
............... 161
D.
1
Continued Fractions
........................................ 163
Appendix
E
The Banach Algebra Property of XOiS
................... 167
Solutions
.......................................................... 169
References
......................................................... 171
List of Symbols
.................................................... 177
Index
............................................................. 179
|
| adam_txt |
Contents
1
Finite
Dimension. 1
1.1 The Lyapunov Center Theorem. 4
1.2 The Weinstein-Moser and Fadell-Rabinowitz
Resonant
Center
Theorems. 7
1.2.1 The Variational Lyapunov-Schmidt
Reduction
. 13
1.2.2
Solution of the Range Equation
. 14
1.2.3
Solution of the Bifurcation Equation
. 15
1.2.4
Proof of the
Weinstein-Moser
Theorem
. 16
1.2.5
Proof of the Fadell-Rabinowitz Theorem
. 20
2
Infinite Dimension
. 29
2.1
The Lyapunov Center Theorem for PDEs
. 31
2.2
Completely Resonant Wave Equations
. 33
2.3
The Case
ρ
Odd
. 35
2.3.1
The Variational Lyapunov-Schmidt Reduction
. 36
2.3.2
The Range Equation
. 37
2.3.3
The Bifurcation Equation
. 41
2.3.4
The Mountain Pass Argument
. 43
2.4
The Case
ρ
Even
. 48
2.5
Multiplicity
. 53
2.6
The Small-Divisor Problem
. 54
3
A Tutorial in Nash-Moser Theory
. 59
3.1
Introduction
. 59
3.2
An Analytic Nash-Moser Theorem
. 60
3.3
A Differentiable Nash-Moser Theorem
. 66
4
Application to the Nonlinear Wave Equation
. 73
4.1
The Zeroth-Order Bifurcation Equation
. 75
4.2
The Finite-Dimensional Reduction
. 77
4.2.1
Solution of the (£>2)-Equation
. 77
4.3
Solution of the Range Equation
. 80
xiv Contents
4.3.1 The Nash-Moser
Scheme
. 83
4.4
Solution of the (^l)-Equation
. 94
4.5
The Linearized Operator
. 98
4.5.1
Decomposition of
C„. 98
4.5.2
Step
1 :
Inversion of
D
. 100
4.5.3
Step
2:
Inversion of Cn
. 104
5
Forced Vibrations
.
Ill
5.1
The Forcing Frequency
ω
є
Q
.
Ill
5.2
The Variational Lyapunov-Schmidt Reduction
. 113
5.2.1
The Range Equation
. 115
5.2.2
The Bifurcation Equation
. 117
5.3
Monotone
ƒ. 119
5.3.1
Step
1:
the L°° Estimate
. 119
5.3.2
Step
2:
the
Я1
Estimate
. 122
5.4
Nonmonotone
ƒ. 124
5.4.1
Step
1 :
the L2k Estimate
. 129
5.4.2
Step
2:
the L°° Estimate
. 130
5.4.3
Step
3:
The
#'
Estimate
. 132
5.4.4
The "Maximum Principle"
. 133
Appendix A Hamiltonian PDEs
. 139
A.
1
The Nonlinear
Schrödinger
Equation
. 139
A.2 The Beam Equation
. 139
A.3 The KdV Equation
. 140
A.4 The
Euler
Equations of Hydrodynamics
. 140
Appendix
В
Critical Point Theory
. 145
B.
1
Preliminaries
. 145
B.2 Minima
. 146
B.3 The Minimax Idea
. 147
B.4 The Mountain Pass Theorem
. 149
Appendix
С
Free Vibrations of Nonlinear Wave Equations: A Global
Result
. 155
Appendix
Đ
Approximation of Irrationals by Rationals
. 161
D.
1
Continued Fractions
. 163
Appendix
E
The Banach Algebra Property of XOiS
. 167
Solutions
. 169
References
. 171
List of Symbols
. 177
Index
. 179 |
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| author_GND | (DE-588)133631656 |
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| id | DE-604.BV023369927 |
| illustrated | Illustrated |
| index_date | 2024-07-02T21:12:14Z |
| indexdate | 2024-07-09T21:17:02Z |
| institution | BVB |
| isbn | 9780817646806 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-016553218 |
| oclc_num | 166362219 |
| open_access_boolean | |
| owner | DE-355 DE-BY-UBR DE-11 |
| owner_facet | DE-355 DE-BY-UBR DE-11 |
| physical | XIV, 180 S. graph. Darst. |
| publishDate | 2007 |
| publishDateSearch | 2007 |
| publishDateSort | 2007 |
| publisher | Birkhäuser |
| record_format | marc |
| series | Progress in non-linear differential equations and their applications |
| series2 | Progress in non-linear differential equations and their applications |
| spelling | Berti, Massimiliano Verfasser (DE-588)133631656 aut Nonlinear oscillations of Hamiltonian PDEs Massimiliano Berti Boston [u.a.] Birkhäuser 2007 XIV, 180 S. graph. Darst. txt rdacontent n rdamedia nc rdacarrier Progress in non-linear differential equations and their applications 74 Opérateur hamiltonien Oscillations non linéaires Systèmes hamiltoniens Équations aux dérivées partielles Differential equations, Partial Hamiltonian operator Hamiltonian systems Nonlinear oscillations Hamiltonsches System (DE-588)4139943-2 gnd rswk-swf Periodische Lösung (DE-588)4199269-6 gnd rswk-swf Nichtlineare Wellengleichung (DE-588)4042104-1 gnd rswk-swf Verzweigung Mathematik (DE-588)4078889-1 gnd rswk-swf Hamiltonsches System (DE-588)4139943-2 s Nichtlineare Wellengleichung (DE-588)4042104-1 s Periodische Lösung (DE-588)4199269-6 s Verzweigung Mathematik (DE-588)4078889-1 s DE-604 Progress in non-linear differential equations and their applications 74 (DE-604)BV007934389 74 Digitalisierung UB Regensburg application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016553218&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Berti, Massimiliano Nonlinear oscillations of Hamiltonian PDEs Progress in non-linear differential equations and their applications Opérateur hamiltonien Oscillations non linéaires Systèmes hamiltoniens Équations aux dérivées partielles Differential equations, Partial Hamiltonian operator Hamiltonian systems Nonlinear oscillations Hamiltonsches System (DE-588)4139943-2 gnd Periodische Lösung (DE-588)4199269-6 gnd Nichtlineare Wellengleichung (DE-588)4042104-1 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| subject_GND | (DE-588)4139943-2 (DE-588)4199269-6 (DE-588)4042104-1 (DE-588)4078889-1 |
| title | Nonlinear oscillations of Hamiltonian PDEs |
| title_auth | Nonlinear oscillations of Hamiltonian PDEs |
| title_exact_search | Nonlinear oscillations of Hamiltonian PDEs |
| title_exact_search_txtP | Nonlinear oscillations of Hamiltonian PDEs |
| title_full | Nonlinear oscillations of Hamiltonian PDEs Massimiliano Berti |
| title_fullStr | Nonlinear oscillations of Hamiltonian PDEs Massimiliano Berti |
| title_full_unstemmed | Nonlinear oscillations of Hamiltonian PDEs Massimiliano Berti |
| title_short | Nonlinear oscillations of Hamiltonian PDEs |
| title_sort | nonlinear oscillations of hamiltonian pdes |
| topic | Opérateur hamiltonien Oscillations non linéaires Systèmes hamiltoniens Équations aux dérivées partielles Differential equations, Partial Hamiltonian operator Hamiltonian systems Nonlinear oscillations Hamiltonsches System (DE-588)4139943-2 gnd Periodische Lösung (DE-588)4199269-6 gnd Nichtlineare Wellengleichung (DE-588)4042104-1 gnd Verzweigung Mathematik (DE-588)4078889-1 gnd |
| topic_facet | Opérateur hamiltonien Oscillations non linéaires Systèmes hamiltoniens Équations aux dérivées partielles Differential equations, Partial Hamiltonian operator Hamiltonian systems Nonlinear oscillations Hamiltonsches System Periodische Lösung Nichtlineare Wellengleichung Verzweigung Mathematik |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=016553218&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV007934389 |
| work_keys_str_mv | AT bertimassimiliano nonlinearoscillationsofhamiltonianpdes |