Partial differential equations and operators

Fundamental solutions and semigroups Part I

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173 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,...

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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.

  1. Distributions, differential operators and examples
    1. Test functions and distributions
    2. Convergence of test functions
    3. Space of test functions
    4. Distributions
    5. Differentiation of distributions
    6. The space (Ω)
    7. Convergence properties of distributions
    8. Supports of distributions
    9. Distributions with compact support
    10. Convolution of a test function and a distribution
    11. Convolution of distributions
    12. Approximate identity
    13. Distributions and C8-diffeomorphisms
    14. Tempered distributions and Fourier transforms
    15. Rapidly decreasing functions
    16. Tempered distributions
    17. Fourier transforms of tempered distributions
    18. Examples of Fourier transforms
    19. Convergence factors
    20. Partial Fourier transformation
  2. Fundamental solutions
    1. Introduction and examples
    2. Hypo-elliptic operators
    3. Ordinary differential equations with constant coefficients
    4. Fundamental solutions of the Cauchy-Riemann operator
    5. Fundamental solutions of the Laplace equation in two dimensions
    6. Fundamental solutions of the heat equation
    7. Fundamental solutions of the Laplace operator in several space dimensions
    8. The free Schr¨odinger equation
  3. Fundamental solutions of the wave operator
    1. Fundamental solutions of the wave operator in one space dimension
    2. Fundamental solutions of the wave equation in several space dimensions
    3. Fundamental solutions which are invariant under certain Lorentz transformations
    4. Explicit formulas for the fundamental solutions
  4. Proofs of some main results
    1. Convolution products: formulation of some results
    2. Proofs
    3. Fourier transform and its inverse
    4. Riesz-Thorin interpolation
    5. Theorem of Malgrange and Ehrenpreis
    6. Sobolev theory
    7. Elliptic operators
    8. Sobolev spaces
    9. Quadratic forms and a compact embedding result
    10. Paley-Wiener theorems
    11. Multiplicative distributions
    12. The representation theorem for the dual of C0pXq
    13. Runge’s theorem
It is very good and useful to scientists and engineers.
8. April 2014 um 23:19
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