Second order partial differential equations in Hilbert spaces / Giuseppe Da Prato & Jerzy Zabczyk.

Format:

Book

Author/Creator:

Da Prato, Giuseppe.

Contributor:

Zabczyk, Jerzy.

Series:

London Mathematical Society lecture note series ; 293.

London Mathematical Society lecture note series ; 293

Language:

English

Subjects (All):

Differential equations, Partial.

Hilbert space.

Physical Description:

xvi, 379 pages : illustrations ; 23 cm.

Place of Publication:

Cambridge, UK ; New York, NY : Cambridge University Press, 2002.

Contents:

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 R[superscript [infinity] 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 UC[subscript b](H) and UC[superscript 1 subscript b](H) 36

2.3.2 Interpolatory estimates 39

2.3.3 Additional interpolation results 42

3 The heat equation 44

3.2 Strict solutions 48

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 [Delta subscript Q]u = 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 [pi]-Convergence 112

6.3.2 Properties of the [pi]-semigroup (R[subscript 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 132

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

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

8.2 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 L[superscript 2] 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 L[superscript 2](H,[mu];H) 193

9.2 Sobolev spaces 194

9.2.1 The space W[superscript 1,2](H,[mu]) 196

9.2.2 Some additional summability results 197

9.2.3 Compactness of the embedding W[superscript 1,2](H,[mu]) [subset or is implied by] L[superscript 2](H,[mu]) 198

9.2.4 The space W[superscript 2,2](H,[mu]) 201

9.3 The Malliavin derivative 203

10 Ornstein-Uhlenbeck semigroups on L[superscript p](H,[mu]) 205

10.1 Extension of (R[subscript t]) to L[superscript p](H,[mu]) 206

10.1.1 The adjoint of (R[subscript t]) in L[superscript 2](H,[mu]) 211

10.2 The infinitesimal generator of (R[subscript t]) 212

10.2.1 Characterization of the domain of L[subscript 2] 215

10.3 The case when (R[subscript t]) is strong Feller 217

10.3.1 Additional regularity properties of (R[subscript t]) 221

10.3.2 Hypercontractivity of (R[subscript t]) 224

10.4 A representation formula for (R[subscript t]) in terms of the second quantization operator 228

10.4.1 The second quantization operator 228

10.4.2 The adjoint of (R[subscript t]) 230

10.5 Poincare and log-Sobolev inequalities 230

10.5.1 The case when M = 1 and Q = I 232

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 (P[subscript t]) in L[superscript p](H,v) 256

11.2.3 The integration by parts formula 260

11.2.4 Existence of a density 263

12 Gradient systems 267

12.1.1 Assumptions and setting of the problem 268

12.1.2 The Sobolev space W[superscript 1,2](H,v) 271

12.1.3 Symmetry of the operator N[subscript 0] 272

12.1.4 The m-dissipativity of N[subscript 1] on L[superscript 1](H,v) 274

12.2 The m-dissipativity of N[subscript 2] on L[superscript 2](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 [varepsilon] [right arrow] 0 314

14 Hamilton-Jacobi inclusions 316

14.2 Excessive weights and an existence result 317

14.3 Weak solutions as value functions 324

14.4 Excessive measures for Wiener processes 328

A Interpolation spaces 335

A.1 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.1 Definition of null controllability 338

B.2 Main results 339

B.3 Minimal energy 340

C Semiconcave functions and Hamilton-Jacobi semigroups 347

C.1 Continuity modulus 347

C.2 Semiconcave and semiconvex functions 348

C.3 The Hamilton-Jacobi semigroups 351.

Notes:

Includes bibliographical references (pages 358-375) and index.

ISBN:

0521777291

OCLC:

49225683

Online:

Publisher description