Second Order Partial Differential Equations in Hilbert Spaces

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

Giuseppe Da Prato (Author), Jerzy Zabczyk (Author)

9780521777292, Cambridge University Press

Paperback, published 25 July 2002

396 pages
22.9 x 15.2 x 2.2 cm, 0.58 kg

'… 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

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.

Part I. Theory in the Space of Continuous Functions: 1. Gaussian measures
2. Spaces of continuous functions
3. Heat equation
4. Poisson's equation
5. Elliptic equations with variable coefficients
6. Ornstein-Uhlenbeck equations
7. General parabolic equations
8. Parabolic equations in open sets
Part II. Theory in Sobolev Spaces with a Gaussian Measure: 9. L2 and Sobolev spaces
10. Ornstein-Uhlenbeck semigroups on Lp(H, mu)
11. Perturbations of Ornstein-Uhlenbeck semigroups
12. Gradient systems
Part II. Applications to Control Theory: 13. Second order Hamilton-Jacobi equations
14. Hamilton-Jacobi inclusions.

Subject Areas: Probability & statistics [PBT]