Second Order Partial Differential Equations in Hilbert Spaces 1st Edition by Giuseppe Da Prato, Jerzy Zabczyk ISBN 978-0521777292 0521777291
Original price was: $50.00.$35.00Current price is: $35.00.
Second Order Partial Differential Equations in Hilbert Spaces 1st Edition by Giuseppe Da Prato, Jerzy Zabczyk – Ebook PDF Instant Download/Delivery: 978-0521777292, 0521777291
Full download Second Order Partial Differential Equations in Hilbert Spaces 1st Edition after payment
Product details:
ISBN 10: 0521777291
ISBN 13: 978-0521777292
Author: Giuseppe Da Prato, Jerzy Zabczyk
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.
Table of contents:
I THEORY IN SPACES OF CONTINUOUS FUNCTIONS
1 Gaussian measures
1.1 Introduction and preliminaries
1.2 Definition and first properties of Gaussian measures
1.2.1 Measures in metric spaces
1.2.2 Gaussian measures
1.2.3 Computation of some Gaussian integrals
1.2.4 The reproducing kernel
1.3 Absolute continuity of Gaussian measures
1.3.1 Equivalence of product measures in R
1.3.2 The Cameron-Martin formula
1.3.3 The Feldman-Hajek theorem
1.4 Brownian motion
2 Spaces of continuous functions
2.1 Preliminary results
2.2 Approximation of continuous functions
2.3 Interpolation spaces
2.3.1 Interpolation between UC(H) and UC(H)
2.3.2 Interpolatory estimates
2.3.3 Additional interpolation results
3 The heat equation
3.1 Preliminaries
3.2 Strict solutions
3.3 Regularity of generalized solutions
3.3.1 Q-derivatives
3.3.2 Q-derivatives of generalized solutions
3.4 Comments on the Gross Laplacian
3.5 The heat semigroup and its generator
4 Poisson’s equation
4.1 Existence and uniqueness results
4.2 Regularity of solutions
4.3 The equation Aqu = g.
4.3.1 The Liouville theorem
5 Elliptic equations with variable coefficients
5.1 Small perturbations
5.2 Large perturbations
6 Ornstein-Uhlenbeck equations
6.1 Existence and uniqueness of strict solutions
6.2 Classical solutions
6.3 The Ornstein-Uhlenbeck semigroup
6.3.1 π-Convergence
6.3.2 Properties of the -semigroup (Re)
6.3.3 The infinitesimal generator
6.4 Elliptic equations
6.4.1 Schauder estimates
6.4.2 The Liouville theorem
6.5 Perturbation results for parabolic equations
6.6 Perturbation results for elliptic equations
7 General parabolic equations
7.1 Implicit function theorems
7.2 Wiener processes and stochastic equations
7.2.1 Infinite dimensional Wiener processes
7.2.2 Stochastic integration
7.3 Dependence of the solutions to stochastic equations on initial
data
7.3.1 Convolution and evaluation maps
7.3.2 Solutions of stochastic equations
7.4 Space and time regularity of the generalized solutions
7.5 Existence
7.6 Uniqueness
7.6.1 Uniqueness for the heat equation
7.6.2 Uniqueness in the general case
7.7 Strong Feller property
8 Parabolic equations in open sets
8.1 Introduction.
8.2 Regularity of the generalized solution
8.3 Existence theorems
8.4 Uniqueness of the solutions
II THEORY IN SOBOLEV SPACES
9 L2 and Sobolev spaces
9.1 Itô-Wiener decomposition
9.1.1 Real Hermite polynomials
9.1.2 Chaos expansions.
9.1.3 The space L² (H, μ; H
).
9.2 Sobolev spaces
9.2.1 The space W12 (H, μ)
9.2.2 Some additional summability results
9.2.3 Compactness of the embedding W¹² (H, µ) CL²(H, μ)
9.2.4 The space W2.2 (H, μ)
9.3 The Malliavin derivative
10 Ornstein-Uhlenbeck semigroups on LP(Η, μ)
10.1 Extension of (R4) to LP(H, µ)
10.1.1 The adjoint of (Re) in L²(H, μ).
10.2 The infinitesimal generator of (R)
10.2.1 Characterization of the domain of L2.
10.3 The case when (R) is strong Feller
10.3.1 Additional regularity properties of (R)
10.3.2 Hypercontractivity of (R).
10.4 A representation formula for (R4) in terms of the second quan-
tization operator
10.4.1 The second quantization operator.
10.4.2 The adjoint of (R).
10.5 Poincaré and log-Sobolev inequalities.
10.5.1 The case when M = 1 and QI
10.5.2 A generalization
10.6 Some additional regularity results when Q and A commute
11 Perturbations of Ornstein-Uhlenbeck semigroups
11.1 Bounded perturbations
11.2 Lipschitz perturbations
11.2.1 Some additional results on the Ornstein-Uhlenbeck
semigroup
11.2.2 The semigroup (Pt) in LP(H, v)
11.2.3 The integration by parts formula
11.2.4 Existence of a density
12 Gradient systems
12.1 General results
12.1.1 Assumptions and setting of the problem
12.1.2 The Sobolev space W1.2 (H,ν).
12.1.3 Symmetry of the operator No.
12.1.4 The m-dissipativity of Ni on L¹ (H, v)..
12.2 The m-dissipativity of Nā on L²(H, v)
12.3 The case when U is convex
12.3.1 Poincaré and log-Sobolev inequalities.
III APPLICATIONS TO CONTROL THEORY
13 Second order Hamilton-Jacobi equations
13.1 Assumptions and setting of the problem
13.2 Hamilton-Jacobi equations with a Lipschitz Hamiltonian.
13.2.1 Stationary Hamilton-Jacobi equations
13.3 Hamilton-Jacobi equation with a quadratic Hamiltonian
13.3.1 Stationary equation
13.4 Solution of the control problem
13.4.1 Finite horizon.
13.4.2 Infinite horizon
13.4.3 The limit as→0.
14 Hamilton-Jacobi inclusions
14.1 Introduction
14.2 Excessive weights and an existence result
14.3 Weak solutions as value functions
14.4 Excessive measures for Wiener processes
IV APPENDICES
A Interpolation spaces
A.1 The interpolation theorem
A.2 Interpolation between a Banach space X and the domain of
a linear operator in X
B Null controllability
B.1 Definition of null controllability
B.2 Main results
B.3 Minimal energy
C Semiconcave functions and Hamilton-Jacobi semigroups
C.1 Continuity modulus
C.2 Semiconcave and semiconvex functions
C.3 The Hamilton-Jacobi semigroups
Bibliography
Index
People also search for:
first order and second order partial differential equations
first order partial differential equations problems and solutions pdf
a second order partial differential equation
first and second order partial differential equations
application of second order partial differential equation
Tags: Giuseppe Da Prato, Jerzy Zabczyk, Second Order, Differential Equations, Hilbert Spaces