Asymptotic analysis for periodic structures:
Gespeichert in:
| Beteiligte Personen: | , , |
|---|---|
| Format: | Buch |
| Sprache: | Englisch |
| Veröffentlicht: |
Amsterdam
North-Holland Publ. Co.
1978
|
| Schriftenreihe: | Studies in mathematics and its applications.
5. |
| Schlagwörter: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001482837&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Umfang: | XXIV, 700 S. |
| ISBN: | 0444851720 |
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| 245 | 1 | 0 | |a Asymptotic analysis for periodic structures |c Alain Bensoussan ; Jacques L. Lions ; George Papanicolaou* |
| 264 | 1 | |a Amsterdam |b North-Holland Publ. Co. |c 1978 | |
| 300 | |a XXIV, 700 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a Studies in mathematics and its applications. |v 5. | |
| 650 | 4 | |a Développements asymptotiques | |
| 650 | 4 | |a Probabilités | |
| 650 | 4 | |a Problèmes aux limites - Solutions numériques | |
| 650 | 4 | |a Équations aux dérivées partielles - Solutions numériques | |
| 650 | 4 | |a Boundary value problems |x Numerical solutions | |
| 650 | 4 | |a Differential equations, Partial |x Asymptotic theory | |
| 650 | 4 | |a Probabilities | |
| 650 | 0 | 7 | |a Asymptotische Entwicklung |0 (DE-588)4112609-9 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Randwertproblem |0 (DE-588)4048395-2 |2 gnd |9 rswk-swf |
| 650 | 0 | 7 | |a Partielle Differentialgleichung |0 (DE-588)4044779-0 |2 gnd |9 rswk-swf |
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| 700 | 1 | |a Lions, Jacques-Louis |d 1928-2001 |e Verfasser |0 (DE-588)124055397 |4 aut | |
| 700 | 1 | |a Papanicolaou, George |d 1943- |e Verfasser |0 (DE-588)171732456 |4 aut | |
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Datensatz im Suchindex
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| adam_text | TABLE OF CONTENTS
Introduction v
Table of Contents xv
Chapter 1 : Elliptic Operators
Orientation 1
1. Setting of the model problem 2
1.1 Setting of the problem (I) 2
1.2 Setting of the problem (II):boundary conditions 4
1.3 An example: a one dimensional problem 8
2. Asymptotic expansions 11
2.1 Orientation 11
2.2 Asymptotic expansions using multiple scales 12
2.3 Remarks on the homogenized operator 16
2.4 Justification of the asymptotic expansion
for Dirichlet s boundary conditions 19
2.5 Higher order terms in the expansion 21
2.6 Extensions 22
3. Energy proof of the homogenization formula 23
3.1 Orientation: statement of the main result 23
3.2 Proof of the convergence theorem 24
3.3 A remark on the use of the adjoint expansion 28
3.4 Comparison results 31
xv
xvi LIST OF CONTENTS
4. Lp estimates 35
4. 1 Estimates for the Dirichlet problem 35
4.2 Reduction of the equation 38
4.3 Proof of Theorem 4.2 40
4.4 Local estimates 44
4.5 Extensions 46
5. Correctors 49
5.1 Orientation 49
5.2 Structure of the first corrector:
Statement of theorem 49
5.3 Proof of Theorem 5.1 53
5.4 First order system and asymptotic expansion 59
5.5 Correctors: Error estimates for Dirichlet s problem .... 65
6. Second order elliptic operators with non uniformly
oscillating coefficients 71
6.1 Setting of the problem and general formulas 71
6.2 Homogenization of transmission problems 77
6.3 Proof of Theorem 6.1 82
6.4 Another approach to Theorem 6.1 84
7. Complements on boundary conditions 87
7.1 A remark on the nonhomogeneous Neumann s problem 87
7.2 Higher order boundary conditions 89
7.3 Proof of (7.6), (7.7) 94
LIST OF CONTENTS xvii
8. Reiterated homogenization 96
8.1 Setting of the problem: Statement of main result 96
8.2 Approximation by smooth coefficients 103
8.3 Asymptotic expansion igg
8.4 Proof of the reiteration formula
for smooth coefficients 112
8.5 Correctors 116
9. Homogenization of elliptic systems 117
9.1 Setting of the problem 117
9.2 Statement of the homogenization procedure 120
9.3 Proof of the homogenization theorem 123
9.4 Correctors 126
10. Homogenization of the Stokes equation 129
10.1 Orientation 129
10.2 Statement of the problem
and the homogenization theorem 129
10.3 Proof of the homogenization theorem 132
10.4 Asymptotic expansion 13g
11. Homogenization of equations of Maxwell s type 138
11.1 Setting of the problem 138
11.2 Asymptotic expansions 140
11.3 Another asymptotic expansion 144
11.4 Compensated compactness 147
11.5 Homogenization theorem 150
11.6 Zero order term 155
11.7 Remark on a regularization method 157
xviii LIST OF CONTENTS
12. Homogenization with rapidly oscillating potentials 158
12.1 Orientation 158
12.2 Asymptotic expansion 159
12.3 Estimates for the spectrum and homogenization 161
12.4 Correctors 167
12.5 Almost periodic potentials 168
12.6 Neumann s problem 170
12.7 Higher order equations 173
12.8 Oscillating potential and oscillatory coefficients 176
12 . 9 A phenomenon of uncoupling 179
13. Study of lower order terms 181
13.1 Orientation 181
13.2 Asymptotic expansion 183
13.3 Energy estimates 185
14. Singular perturbations and homogenization 188
14.1 Orientation 188
14.2 Asymptotic expansion 189
14.3 Homogenization with respect to A 191
15. Non local limits 194
15.1 Setting of the problem 194
15 . 2 Non local homogenized operator 196
15.3 Homogenization theorem 199
16. Introduction to non linear problems 200
16.1 Formal general formulas 200
16.2 Compact perturbations 202
16.3 Non compact perturbations 203
16.4 Non linearities in the higher derivatives 206
LIST OF CONTENTS xix
17. Homogenization of multi valued operators 207
17.1 Orientation 207
17.2 A formal procedure for the homogenization
of problems of the calculus of variations 209
17.3 Unilateral variational inequalities 214
18. Comments and problems 218
Bibliography 228
Chapter 2 : Evolution Operators
Orientation 233
1. Parabolic operators: Asymptotic expansions 234
1.1 Notations and orientation 234
1.2 Variational formulation 235
1.3 Asymptotic expansions: Preliminary formulas 243
1.4 Asymptotic expansions: The case k = 1 245
1.5 Asymptotic expansions: The case k = 2 246
1.6 Asymptotic expansions: The case k = 3 248
1.7 Other form of homogenization formulas 250
1.8 The role of k 252
2. Convergence of the homogenization
of parabolic equations 253
2.1 Statement of the homogenization result 253
2.2 Proof of the homogenization when k = 2 253
2.3 Reduction to the smooth case 257
2.4 Proof of the homogenization when 0 k 2 259
2.5 Proof of the homogenization when k 2 266
2.6 Proof of the homogenization formulas
when a. . 6 lT (Rn x R ) using LP estimates 269
2.7 The Lp estimates 271
2.8 The adjoint expansion 276
XX LIST OF CONTENTS
2.9 Use of the maximum principle 277
2.10 Higher order equations and systems 278
2. 11 Correctors 282
2.12 Non linear problems 287
2.13 Remarks on averaging 293
3. Evolution operators of hyperbolic, Petrowsky
or Schrodinger type 299
3.1 Orientation 299
3.2 Linear operators with coefficients
which are regular in t 300
3.3 Linear operators with coefficients
which are irregular in t 304
3.4 Asymptotic expansions (I) 306
3.5 Asymptotic expansions (II) 308
3.6 Remarks on correctors 312
3.7 Remarks on nonlinear problems 314
3.8 Remarks on Schrodinger type equations 317
3.9 Nonlocal operators 318
4. Comments and problems 325
Bibliography 342
Chapter 3: Probabilistic Problems and Methods
Orientation 345
1. Stochastic differential equations
and connections with partial differential equations .. 348
1.1 Stochastic integrals 348
1.2 Ito s formula 350
1.3 Strong formulation of stochastic differential
equations 351
1.4 Connections with partial differential equations 353
LIST OF CONTENTS xxi
2. Martingale formulation of stochastic
differential equations 357
2.1 Martingale problem 357
2.2 Weak formulation of stochastic differential equations .. 359
2.3 Connections with P. D. E 361
3. Some results from ergodic theory 364
3.1 General results 364
3.2 Ergodic properties of diffusions on the torus 370
3.3 Invariant measure and the Fredholm alternative 377
4. Homogenization with a constant coefficients
limit operator 383
4.1 Diffusion without drift 384
4.2 Diffusion with unbounded drift 399
4.3 Convergence of functionals
and probabilistic proof of homogenization 405
5. Analytic approach to the problem (4.76) 414
5.1 The method of asymptotic expansions 414
5.2 The method of energy 419
6. Operators with locally periodic coefficients 429
6.1 Setting of the problem 429
6.2 Probabilistic approach 431
6.3 Remarks on the martingale approach
and the adjoint expansion method 442
6.4 Analytic approach to problem 6.5 444
7. Reiterated homogenization 455
7.1 Setting of the problem 455
7.2 Proof of Theorem 7.1 463
xxii LIST OF CONTENTS
8 . Problems with potentials 467
8.1 A variant of Theorem 6.3 467
8.2 A general problem with potentials 470
9. Homogenization of reflected diffusion processes 475
9 .1 Setting of the problem 475
9.2 Proof of convergence 478
9.3 Applications to partial differential equations 483
10. Evolution problems 488
10.1 Notation and setting of problems 488
10.2 Fredholm alternative for an evolution operator 489
10.3 Case k 2 496
10.4 Case k = 2 505
10.5 Case k 2 509
10.6 Applications to parabolic equations 515
11. Averaging 516
11.1 Setting of the problem 516
11.2 Proof of Theorem 11.1 518
11.3 Remarks on generalized averaging 525
12. Comments and problems 529
Bibliography 534
Chapter 4 : High Frequency wave Propagation in Periodic Structures.
Orientation 537
1. Formulation of the problems 537
1.1 High frequency wave propagation 538
1.2 Propagation in periodic structures 545
LIST OF CONTENTS xxiii
2. The W. K. B. or geometrical optics method
2.1 Expansion for the Klein Gordon equation 547
2. 2 Eiconal equation and rays 550
2.3 Transport equations 552
2.4 Connections with the static problem 556
2.5 Propagation of energy 557
2.6 Spatially localized data 559
2.7 Expansion for the fundamental solution 563
2.8 Expansion near smooth caustics 565
2.9 Impact problem 566
2.10 Symmetric hyperbolic systems 568
2.11 Expansions for symmetric hyperbolic systems
(low frequency) 574
2.12 Expansion for symmetric hyperbolic systems
(probabilistic) 531
2.13 Expansion for symmetric hyperbolic systems
(high frequency) 587
2.14 W. K. B. for dissipative symmetric hyperbolic systems .. 600
2.15 Operator form of the W. K. B 609
3. Spectral theory for differential operators
with periodic coefficients 614
3.1 The shifted cell problem for a second order
elliptic operator 614
3.2 The Bloch expansion theorem 616
3.3 Bloch expansion for the acoustic equations 618
3.4 Bloch expansion for Maxwell s equations 619
3. 5 The dynamo problem 620
3.6 Some nonselfadjoint problems 621
xxiv LIST OF CONTENTS
4. Simple applications of the spectral expansion
4.1 Lattice waves 627
4. 2 Schrodinger s equation 629
4.3 Nature of the expansion 632
4.4 Connections with the static theory 637
4.5 Validity of the expansion 640
4.6 Relation between the Hilbert
and Chapman Enskog expansion 645
4.7 Spatially localized data and stationary phase 646
4.8 Behavior of probability amplitudes 649
4.9 The acoustic equations 650
4.10 Dual homogenization formulas 654
4.11 Maxwell s equations 661
4.12 A one dimensional example 667
5. The general geometrical optics expansion
5.1 Expansion for Schrodinger s equation 671
5.2 Eiconal equations and rays 677
5.3 Transport equations 678
5.4 Connections with the static theory 632
5.5 Spatially localized data 684
5.6 Behavior of probability amplitudes 685
5. 7 Expansion for the wave equation 685
5.8 Expansion for the heat equation 687
6. Comments and problems 694
Bibliography 696
|
| any_adam_object | 1 |
| author | Bensoussan, Alain 1940- Lions, Jacques-Louis 1928-2001 Papanicolaou, George 1943- |
| author_GND | (DE-588)13295947X (DE-588)124055397 (DE-588)171732456 |
| author_facet | Bensoussan, Alain 1940- Lions, Jacques-Louis 1928-2001 Papanicolaou, George 1943- |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.35 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
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| id | DE-604.BV002256530 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T07:48:25Z |
| institution | BVB |
| isbn | 0444851720 |
| language | English |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-001482837 |
| oclc_num | 3844544 |
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| physical | XXIV, 700 S. |
| publishDate | 1978 |
| publishDateSearch | 1978 |
| publishDateSort | 1978 |
| publisher | North-Holland Publ. Co. |
| record_format | marc |
| series | Studies in mathematics and its applications. |
| series2 | Studies in mathematics and its applications. |
| spellingShingle | Bensoussan, Alain 1940- Lions, Jacques-Louis 1928-2001 Papanicolaou, George 1943- Asymptotic analysis for periodic structures Studies in mathematics and its applications. Développements asymptotiques Probabilités Problèmes aux limites - Solutions numériques Équations aux dérivées partielles - Solutions numériques Boundary value problems Numerical solutions Differential equations, Partial Asymptotic theory Probabilities Asymptotische Entwicklung (DE-588)4112609-9 gnd Randwertproblem (DE-588)4048395-2 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| subject_GND | (DE-588)4112609-9 (DE-588)4048395-2 (DE-588)4044779-0 (DE-588)4128130-5 |
| title | Asymptotic analysis for periodic structures |
| title_auth | Asymptotic analysis for periodic structures |
| title_exact_search | Asymptotic analysis for periodic structures |
| title_full | Asymptotic analysis for periodic structures Alain Bensoussan ; Jacques L. Lions ; George Papanicolaou* |
| title_fullStr | Asymptotic analysis for periodic structures Alain Bensoussan ; Jacques L. Lions ; George Papanicolaou* |
| title_full_unstemmed | Asymptotic analysis for periodic structures Alain Bensoussan ; Jacques L. Lions ; George Papanicolaou* |
| title_short | Asymptotic analysis for periodic structures |
| title_sort | asymptotic analysis for periodic structures |
| topic | Développements asymptotiques Probabilités Problèmes aux limites - Solutions numériques Équations aux dérivées partielles - Solutions numériques Boundary value problems Numerical solutions Differential equations, Partial Asymptotic theory Probabilities Asymptotische Entwicklung (DE-588)4112609-9 gnd Randwertproblem (DE-588)4048395-2 gnd Partielle Differentialgleichung (DE-588)4044779-0 gnd Numerisches Verfahren (DE-588)4128130-5 gnd |
| topic_facet | Développements asymptotiques Probabilités Problèmes aux limites - Solutions numériques Équations aux dérivées partielles - Solutions numériques Boundary value problems Numerical solutions Differential equations, Partial Asymptotic theory Probabilities Asymptotische Entwicklung Randwertproblem Partielle Differentialgleichung Numerisches Verfahren |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=001482837&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000646 |
| work_keys_str_mv | AT bensoussanalain asymptoticanalysisforperiodicstructures AT lionsjacqueslouis asymptoticanalysisforperiodicstructures AT papanicolaougeorge asymptoticanalysisforperiodicstructures |
Inhaltsverzeichnis
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Teilbibliothek Mathematik & Informatik
| Signatur: |
0102 MAT 667f 2001 A 7341
Lageplan |
|---|---|
| Exemplar 1 | Ausleihbar Am Standort |