Partial differential equations and operators

Fundamental solutions and semigroups Part II
Rezension :
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249 pages
Sprache:
 English
In this book, which is basically self-contained, we concentrate on partial differential equations in mathematical physics and on operator semigroups with their generators.
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Über den Autor

Since 2009 the author is retired from the University of Antwerp. Until the present day his teaching duties include a course on ``Partial Differential Equations and Operators’’ and one on ``Advanced Stochastic Processes’’. In the sixties the author was a student at the Catholic University of Nijmegen,...

Description
Content

In this book, which is basically self-contained, we concentrate on partial differential equations in mathematical physics and on operator semigroups with their generators. A central theme is a thorough treatment of distribution theory. This is done via convolution products, Fourier transforms, and fundamental solutions of partial differential operators with constant coefficients. Linear initial value problems are treated via operator semigroups. A relationship between so-called Feller-Dynkin semigroups and Markov processes is described. Finally, Feynman-Kac semigroups are introduced.

To see Section 1–4 download Partial differential equations and operators: Part I

  1. Operators in Hilbert space
    1. Some results in Banach algebras
    2. Symbolic calculus
    3. On square roots in Banach algebras
    4. On C*-algebras
    5. On Gelfand transforms
    6. Resolution of the identity
    7. Closed linear operators
  2. Operator semigroups and Markov processes
    1. Generalities on semigroups
    2. Examples
    3. Uniformly continuous semigroups
    4. Self-adjoint semigroups
    5. Translation group
    6. Gaussian semigroup
    7. Wave operator
    8. Adjoint semigroups
    9. Dyson-Phillips expansion
    10. Stone’s theorem
    11. Convolution semigroups of measures
    12. Semigroups acting on operators
    13. Quantum dynamical semigroups
    14. Semigroups for system theory
    15. Semigroups and pseudo-differential operators
    16. Quadratic forms and semigroups
    17. Ornstein-Uhlenbeck semigroup
    18. Evolutions and semigroups
    19. Markov processes
    20. Feynman-Kac semigroups
    21. KMS formula
    22. Harmonic functions on a strip
  3. Holomorphic semigroups
    1. Introduction
    2. Exponentially bounded analytic semigroups
    3. Bounded analytic semigroups
    4. Bounded analytic semigroups and the Crank-Nicolson iteration scheme
    5. Stability of the Crank-Nicolson iteration scheme
  4. Elements of functional analysis
    1. Theorem of Hahn-Banach
    2. Baire category
    3. Banach-Steinhaus theorems: barreled spaces
    4. The open mapping theorem
    5. Krein-Smulian and the Eberlein-Smulian theorem