Second order partial differential equations in Hilbert spaces:
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| Main Authors: | , |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
Cambridge [u.a.]
Cambridge Univ. Press
2002
|
| Edition: | 1. publ. |
| Series: | London Mathematical Society lecture note series
293 |
| Subjects: | |
| Links: | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009855489&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| Physical Description: | XVI, 379 S. |
| ISBN: | 0521777291 |
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| 100 | 1 | |a Da Prato, Giuseppe |d 1936-2023 |e Verfasser |0 (DE-588)121352641 |4 aut | |
| 245 | 1 | 0 | |a Second order partial differential equations in Hilbert spaces |c Giuseppe Da Prato ; Jerzy Zabczyk |
| 250 | |a 1. publ. | ||
| 264 | 1 | |a Cambridge [u.a.] |b Cambridge Univ. Press |c 2002 | |
| 300 | |a XVI, 379 S. | ||
| 336 | |b txt |2 rdacontent | ||
| 337 | |b n |2 rdamedia | ||
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| 490 | 1 | |a London Mathematical Society lecture note series |v 293 | |
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| 650 | 7 | |a Hilbertruimten |2 gtt | |
| 650 | 4 | |a Équations aux dérivées partielles | |
| 650 | 4 | |a Differential equations, Partial | |
| 650 | 4 | |a Hilbert space | |
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Record in the Search Index
| DE-BY-TUM_call_number | 0202 MAT 354f 2003 A 145 |
|---|---|
| DE-BY-TUM_katkey | 1395088 |
| DE-BY-TUM_location | 02 |
| DE-BY-TUM_media_number | 040020712421 |
| _version_ | 1821939278336229377 |
| adam_text | Contents
Preface x
I THEORY IN SPACES OF CONTINUOUS
FUNCTIONS 1
1 Gaussian measures 3
1.1 Introduction and preliminaries 3
1.2 Definition and first properties of Gaussian measures 7
1.2.1 Measures in metric spaces 7
1.2.2 Gaussian measures 8
1.2.3 Computation of some Gaussian integrals 11
1.2.4 The reproducing kernel 12
1.3 Absolute continuity of Gaussian measures 17
1.3.1 Equivalence of product measures in K°° 18
1.3.2 The Cameron Martin formula 22
1.3.3 The Feldman Hajek theorem 24
1.4 Brownian motion 27
2 Spaces of continuous functions 30
2.1 Preliminary results 30
2.2 Approximation of continuous functions 33
2.3 Interpolation spaces 36
2.3.1 Interpolation between UCb(H) and UCl(H) 36
2.3.2 Interpolators estimates 39
2.3.3 Additional interpolation results 42
3 The heat equation 44
3.1 Preliminaries 44
3.2 Strict solutions 48
v
vi Contents
3.3 Regularity of generalized solutions 54
3.3.1 Q derivatives 54
3.3.2 Q derivatives of generalized solutions 57
3.4 Comments on the Gross Laplacian 67
3.5 The heat semigroup and its generator 69
4 Poisson s equation 76
4.1 Existence and uniqueness results 76
4.2 Regularity of solutions 78
4.3 The equation Aqu = g 83
4.3.1 The Liouville theorem 87
5 Elliptic equations with variable coefficients 90
5.1 Small perturbations 90
5.2 Large perturbations 93
6 Ornstein Uhlenbeck equations 99
6.1 Existence and uniqueness of strict solutions 100
6.2 Classical solutions 103
6.3 The Ornstein Uhlenbeck semigroup 111
6.3.1 7r Convergence 112
6.3.2 Properties of the 7r semigroup (i?t) 113
6.3.3 The infinitesimal generator 114
6.4 Elliptic equations 116
6.4.1 Schauder estimates 119
6.4.2 The Liouville theorem 121
6.5 Perturbation results for parabolic equations 122
6.6 Perturbation results for elliptic equations 124
7 General parabolic equations 127
7.1 Implicit function theorems 128
7.2 Wiener processes and stochastic equations 131
7.2.1 Infinite dimensional Wiener processes 131
7.2.2 Stochastic integration l32
7.3 Dependence of the solutions to stochastic equations on initial data 133
7.3.1 Convolution and evaluation maps 133
7.3.2 Solutions of stochastic equations 138
7.4 Space and time regularity of the generalized solutions 139
7.5 Existence 142
Contents vii
7.6 Uniqueness 144
7.6.1 Uniqueness for the heat equation 145
7.6.2 Uniqueness in the general case 146
7.7 Strong Feller property 150
8 Parabolic equations in open sets 156
8.1 Introduction 156
8.2 Regularity of the generalized solution 158
8.3 Existence theorems 165
8.4 Uniqueness of the solutions 178
II THEORY IN SOBOLEV SPACES 185
9 L2 and Sobolev spaces 187
9.1 Ito Wiener decomposition 188
9.1.1 Real Hermite polynomials 188
9.1.2 Chaos expansions 190
9.1.3 The space L2(H,fj,;H) 193
9.2 Sobolev spaces 194
9.2.1 The space Wl 2(H,n) 196
9.2.2 Some additional summability results 197
9.2.3 Compactness of the embedding WX 2(H, y) C L2(H, /x) 198
9.2.4 The space W2 2(H,fi) 201
9.3 The Malliavin derivative 203
10 Ornstein Uhlenbeck semigroups on Lp(H,y) 205
10.1 Extension of (Rt) to Lp(H,n) 206
10.1.1 The adjoint of (Rt) in L2(H,n) 211
10.2 The infinitesimal generator of (Rt) 212
10.2.1 Characterization of the domain of L2 215
10.3 The case when (Rt) is strong Feller 217
10.3.1 Additional regularity properties of (Rt) 221
10.3.2 Hypercontractivity of (Rt) 224
10.4 A representation formula for (Rt) in terms of the second quan¬
tization operator 228
10.4.1 The second quantization operator 228
10.4.2 The adjoint of (Rt) 230
10.5 Poincare and log Sobolev inequalities 230
10.5.1 The case when M = 1 and Q = / 232
viii Contents
10.5.2 A generalization 235
10.6 Some additional regularity results when Q and A commute . 236
11 Perturbations of Ornstein Uhlenbeck semigroups 238
11.1 Bounded perturbations 239
11.2 Lipschitz perturbations 245
11.2.1 Some additional results on the Ornstein Uhlenbeck
semigroup 251
11.2.2 The semigroup (Pt) in U {H,v) 256
11.2.3 The integration by parts formula 260
11.2.4 Existence of a density 263
12 Gradient systems 267
12.1 General results 268
12.1.1 Assumptions and setting of the problem 268
12.1.2 The Sobolev space Wl 2(H,v) 271
12.1.3 Symmetry of the operator Nq 272
12.1.4 The m dissipativity of TVi oni (/f,i/) 274
12.2 The m dissipativity of N2 on L2(H,v) 277
12.3 The case when U is convex 281
12.3.1 Poincare and log Sobolev inequalities 288
III APPLICATIONS TO CONTROL THEORY 291
13 Second order Hamilton Jacobi equations 293
13.1 Assumptions and setting of the problem 296
13.2 Hamilton Jacobi equations with a Lipschitz Hamiltonian . . . 300
13.2.1 Stationary Hamilton Jacobi equations 302
13.3 Hamilton Jacobi equation with a quadratic Hamiltonian . . . 305
13.3.1 Stationary equation 308
13.4 Solution of the control problem 310
13.4.1 Finite horizon 310
13.4.2 Infinite horizon 312
13.4.3 The limit as e 0 314
14 Hamilton Jacobi inclusions 316
14.1 Introduction 316
14.2 Excessive weights and an existence result 317
14.3 Weak solutions as value functions 324
Contents ix
14.4 Excessive measures for Wiener processes 328
IV APPENDICES 333
A Interpolation spaces 335
A.I The interpolation theorem 335
A.2 Interpolation between a Banach space X and the domain of
a linear operator in X 336
B Null controllability 338
B.I Definition of null controllability 338
B.2 Main results 339
B.3 Minimal energy 340
C Semiconcave functions and Hamilton Jacobi semigroups 347
C.I Continuity modulus 347
C.2 Semiconcave and semiconvex functions 348
C.3 The Hamilton Jacobi semigroups 351
Bibliography 358
Index 376
|
| any_adam_object | 1 |
| author | Da Prato, Giuseppe 1936-2023 Zabczyk, Jerzy 1941- |
| author_GND | (DE-588)121352641 (DE-588)12135234X |
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| classification_tum | MAT 354f |
| ctrlnum | (OCoLC)49225683 (DE-599)BVBBV014403464 |
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| dewey-ones | 515 - Analysis |
| dewey-raw | 515/.353 |
| dewey-search | 515/.353 |
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| dewey-tens | 510 - Mathematics |
| discipline | Mathematik |
| edition | 1. publ. |
| format | Book |
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| id | DE-604.BV014403464 |
| illustrated | Not Illustrated |
| indexdate | 2024-12-20T11:03:56Z |
| institution | BVB |
| isbn | 0521777291 |
| language | English |
| lccn | 2002022269 |
| oai_aleph_id | oai:aleph.bib-bvb.de:BVB01-009855489 |
| oclc_num | 49225683 |
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| owner_facet | DE-703 DE-91G DE-BY-TUM DE-355 DE-BY-UBR DE-188 DE-83 |
| physical | XVI, 379 S. |
| publishDate | 2002 |
| publishDateSearch | 2002 |
| publishDateSort | 2002 |
| publisher | Cambridge Univ. Press |
| record_format | marc |
| series | London Mathematical Society lecture note series |
| series2 | London Mathematical Society lecture note series |
| spellingShingle | Da Prato, Giuseppe 1936-2023 Zabczyk, Jerzy 1941- Second order partial differential equations in Hilbert spaces London Mathematical Society lecture note series Differentiaalvergelijkingen gtt Equações diferenciais parciais de 2ð ordem larpcal Hilbert, Espace de Hilbertruimten gtt Équations aux dérivées partielles Differential equations, Partial Hilbert space Partielle Differentialgleichung (DE-588)4044779-0 gnd Hilbert-Raum (DE-588)4159850-7 gnd Ordnung 2 (DE-588)4350619-7 gnd |
| subject_GND | (DE-588)4044779-0 (DE-588)4159850-7 (DE-588)4350619-7 |
| title | Second order partial differential equations in Hilbert spaces |
| title_auth | Second order partial differential equations in Hilbert spaces |
| title_exact_search | Second order partial differential equations in Hilbert spaces |
| title_full | Second order partial differential equations in Hilbert spaces Giuseppe Da Prato ; Jerzy Zabczyk |
| title_fullStr | Second order partial differential equations in Hilbert spaces Giuseppe Da Prato ; Jerzy Zabczyk |
| title_full_unstemmed | Second order partial differential equations in Hilbert spaces Giuseppe Da Prato ; Jerzy Zabczyk |
| title_short | Second order partial differential equations in Hilbert spaces |
| title_sort | second order partial differential equations in hilbert spaces |
| topic | Differentiaalvergelijkingen gtt Equações diferenciais parciais de 2ð ordem larpcal Hilbert, Espace de Hilbertruimten gtt Équations aux dérivées partielles Differential equations, Partial Hilbert space Partielle Differentialgleichung (DE-588)4044779-0 gnd Hilbert-Raum (DE-588)4159850-7 gnd Ordnung 2 (DE-588)4350619-7 gnd |
| topic_facet | Differentiaalvergelijkingen Equações diferenciais parciais de 2ð ordem Hilbert, Espace de Hilbertruimten Équations aux dérivées partielles Differential equations, Partial Hilbert space Partielle Differentialgleichung Hilbert-Raum Ordnung 2 |
| url | http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=009855489&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA |
| volume_link | (DE-604)BV000000130 |
| work_keys_str_mv | AT dapratogiuseppe secondorderpartialdifferentialequationsinhilbertspaces AT zabczykjerzy secondorderpartialdifferentialequationsinhilbertspaces |
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0202 MAT 354f 2003 A 145
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