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E-raamat: Second Order Partial Differential Equations in Hilbert Spaces

(Polish Academy of Sciences), (Scuola Normale Superiore, Pisa)
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Second Order Partial Differential Equations in Hilbert SpacesSuurem pilt
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Da Prato (Scuola Normale Superiore di Pisa) and Zabczyk (Polish Academy of Sciences, Warsaw) present the state of the rapidly evolving art of the theory of parabolic/elliptic equations in an infinite dimensional Hilbert space theory relevant to functional analysis, mathematical physics, and applied mathematics. The theoretical section covers spaces of continuous functions and Sobolev spaces. Applications to control theory are then discussed. Appendices address interpolation spaces, null controllability, and semiconcave functions and Hamilton-Jacobi semigroups. Annotation c. Book News, Inc., Portland, OR (booknews.com)

Second order linear parabolic and elliptic equations arise frequently in mathematical physics, biology and finance. Here the authors present a state of the art treatment of the subject from a new perspective. They then go on to discuss how the results in the book can be applied to control theory. This area is developing rapidly and there are numerous notes and references that point the reader to more specialized results not covered in the book. Coverage of some essential background material helps to make the book self contained.

State of the art treatment of the subject with background and references for further reading.

Arvustused

' can be warmly recommended to anyone interested in the field.' European Mathematical Society Newsletter ' the authors five an almost optimal presentation: making the central ideas clear, not hiding the problems and the technical efforts needed to overcome them, and discussing in an appropriate way relations to existing work. it will be of enormous help to experts of PhD students starting to work in the field. The LMS Lecture Note Series has the aim of providing 'volumes [ that] are short monographs giving authoritative accounts of the present state of knowledge on a topic of general interest'. The authors have done a fine job, and have successfully achieved this goal.' Niels Jacob, University of Wales, Swansea

Muu info

State of the art treatment of the subject with background and references for further reading.
Preface x
I THEORY IN SPACES OF CONTINUOUS FUNCTIONS 1(184)
Gaussian measures
3(27)
Introduction and preliminaries
3(4)
Definition and first properties of Gaussian measures
7(10)
Measures in metric spaces
7(1)
Gaussian measures
8(3)
Computation of some Gaussian integrals
11(1)
The reproducing kernel
12(5)
Absolute continuity of Gaussian measures
17(10)
Equivalence of product measures in R∞
18(4)
The Cameron-Martin formula
22(2)
The Feldman-Hajek theorem
24(3)
Brownian motion
27(3)
Spaces of continuous functions
30(14)
Preliminary results
30(3)
Approximation of continuous functions
33(3)
Interpolation spaces
36(8)
Interpolation between UCb(H) and UC1b(H)
36(3)
Interpolatory estimates
39(3)
Additional interpolation results
42(2)
The heat equation
44(32)
Preliminaries
44(4)
Strict solutions
48(6)
Regularity of generalized solutions
54(13)
Q-derivatives
54(3)
Q-derivatives of generalized solutions
57(10)
Comments on the Gross Laplacian
67(2)
The heat semigroup and its generator
69(7)
Poisson's equation
76(14)
Existence and uniqueness results
76(2)
Regularity of solutions
78(5)
The equation ΔQu = g
83(7)
The Liouville theorem
87(3)
Elliptic equations with variable coefficients
90(9)
Small perturbations
90(3)
Large perturbations
93(6)
Ornstein-Uhlenbeck equations
99(28)
Existence and uniqueness of strict solutions
100(3)
Classical solutions
103(8)
The Ornstein-Uhlenbeck semigroup
111(5)
π-Convergence
112(1)
Properties of the π-semigroup (Rt)
113(1)
The infinitesimal generator
114(2)
Elliptic equations
116(6)
Schauder estimates
119(2)
The Liouville theorem
121(1)
Perturbation results for parabolic equations
122(2)
Perturbation results for elliptic equations
124(3)
General parabolic equations
127(29)
Implicit function theorems
128(3)
Wiener processes and stochastic equations
131(2)
Infinite dimensional Wiener processes
131(1)
Stochastic integration
132(1)
Dependence of the solutions to stochastic equations on initial data
133(6)
Convolution and evaluation maps
133(5)
Solutions of stochastic equations
138(1)
Space and time regularity of the generalized solutions
139(3)
Existence
142(2)
Uniqueness
144(6)
Uniqueness for the heat equation
145(1)
Uniqueness in the general case
146(4)
Strong Feller property
150(6)
Parabolic equations in open sets
156(29)
Introduction
156(2)
Regularity of the generalized solution
158(7)
Existence theorems
165(13)
Uniqueness of the solutions
178(7)
II THEORY IN SOBOLEV SPACES 185(106)
L2 and Sobolev spaces
187(18)
Ito-Wiener decomposition
188(6)
Real Hermite polynomials
188(2)
Chaos expansions
190(3)
The space L2(H, μ H)
193(1)
Sobolev spaces
194(9)
The space W1,2(H,μ)
196(1)
Some additional summability results
197(1)
Compactness of the embedding W1,2(H, μ) C L2(H, μ)
198(3)
The space W2,2(H, μ)
201(2)
The Malliavin derivative
203(2)
Ornstein-Uhlenbeck semigroups on Lp(H, μ)
205(33)
Extension of (Rt) to Lp(H, μ)
206(6)
The adjoint of (Rt) in L2(H, μ)
211(1)
The infinitesimal generator of (Rt)
212(5)
Characterization of the domain of L2
215(2)
The case when (Rt) is strong Feller
217(11)
Additional regularity properties of (Rt)
221(3)
Hypercontractivity of (Rt)
224(4)
A representation formula for (Rt) in terms of the second quantization operator
228(2)
The second quantization operator
228(2)
The adjoint of (Rt)
230(1)
Poincare and log-Sobolev inequalities
230(6)
The case when M = 1 and Q = I
232(3)
A generalization
235(1)
Some additional regularity results when Q and A commute
236(2)
Perturbations of Ornstein-Uhlenbeck semigroups
238(29)
Bounded perturbations
239(6)
Lipschitz perturbations
245(22)
Some additional results on the Ornstein-Uhlenbeck semigroup
251(5)
The semigroup (Pt) in Lp(H, v)
256(4)
The integration by parts formula
260(3)
Existence of a density
263(4)
Gradient systems
267(24)
General results
268(9)
Assumptions and setting of the problem
268(3)
The Sobolev space W1,2(H, v)
271(1)
Symmetry of the operator N0
272(2)
The m-dissipativity of N1 on L1(H, v)
274(3)
The m-dissipativity of N2 on L2(H, v)
277(4)
The case when U is convex
281(10)
Poincare and log-Sobolev inequalities
288(3)
III APPLICATIONS TO CONTROL THEORY 291(42)
Second order Hamilton-Jacobi equations
293(23)
Assumptions and setting of the problem
296(4)
Hamilton-Jacobi equations with a Lipschitz Hamiltonian
300(5)
Stationary Hamilton-Jacobi equations
302(3)
Hamilton-Jacobi equation with a quadratic Hamiltonian
305(5)
Stationary equation
308(2)
Solution of the control problem
310(6)
Finite horizon
310(2)
Infinite horizon
312(2)
The limit as ε → 0
314(2)
Hamilton-Jacobi inclusions
316(17)
Introduction
316(1)
Excessive weights and an existence result
317(7)
Weak solutions as value functions
324(4)
Excessive measures for Wiener processes
328(5)
IV APPENDICES 333(25)
A Interpolation spaces
335(3)
A.1 The interpolation theorem
335(1)
A.2 Interpolation between a Banach space X and the domain of a linear operator in X
336(2)
B Null controllability
338(9)
B.1 Definition of null controllability
338(1)
B.2 Main results
339(1)
B.3 Minimal energy
340(7)
C Semiconcave functions and Hamilton-Jacobi semigroups
347(11)
C.1 Continuity modulus
347(1)
C.2 Semiconcave and semiconvex functions
348(3)
C.3 The Hamilton-Jacobi semigroups
351(7)
Bibliography 358(18)
Index 376


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