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

Format:

Book

Author/Creator:

Da Prato, Giuseppe, author.

Zabczyk, Jerzy, author.

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:

1 online resource (xvi, 379 pages) : digital, PDF file(s).

Edition:

1st ed.

Place of Publication:

Cambridge : Cambridge University Press, 2002.

Language Note:

English

Summary:

Second order linear parabolic and elliptic equations arise frequently in mathematics and other disciplines. For example parabolic equations are to be found in statistical mechanics and solid state theory, their infinite dimensional counterparts are important in fluid mechanics, mathematical finance and population biology, whereas nonlinear parabolic equations arise in control theory. Here the authors present a state of the art treatment of the subject from a new perspective. The main tools used are probability measures in Hilbert and Banach spaces and stochastic evolution equations. There is then a discussion of how the results in the book can be applied to control theory. This area is developing very rapidly and there are numerous notes and references that point the reader to more specialised results not covered in the book. Coverage of some essential background material will help make the book self-contained and increase its appeal to those entering the subject.

Contents:

Theory in Spaces of Continuous Functions

Gaussian measures

Introduction and preliminaries

Definition and first properties of Gaussian measures

Measures in metric spaces

Computation of some Gaussian integrals

The reproducing kernel

Absolute continuity of Gaussian measures

Equivalence of product measures in R[superscript [infinity]

The Cameron-Martin formula

The Feldman-Hajek theorem

Brownian motion

Spaces of continuous functions

Preliminary results

Approximation of continuous functions

Interpolation spaces

Interpolation between UC[subscript b](H) and UC[superscript 1 subscript b](H)

Interpolatory estimates

Additional interpolation results

The heat equation

Strict solutions

Regularity of generalized solutions

Q-derivatives

Q-derivatives of generalized solutions

Comments on the Gross Laplacian

The heat semigroup and its generator

Poisson's equation

Existence and uniqueness results

Regularity of solutions

The equation [Delta subscript Q]u = g

The Liouville theorem

Elliptic equations with variable coefficients

Small perturbations

Large perturbations

Ornstein-Uhlenbeck equations

Existence and uniqueness of strict solutions

Classical solutions

The Ornstein-Uhlenbeck semigroup

[pi]-Convergence

Properties of the [pi]-semigroup (R[subscript t])

The infinitesimal generator

Elliptic equations

Schauder estimates

Perturbation results for parabolic equations.

Notes:

Title from publisher's bibliographic system (viewed on 05 Oct 2015).

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

ISBN:

1-107-11991-X

1-280-42958-5

9786610429585

0-511-17727-5

0-511-15823-8

0-511-32567-3

0-511-54321-2

0-511-04995-1

OCLC:

56125251