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.
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Partial differential equations and operators
Fundamental solutions and semigroups: Part I
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- Preface
- Chapter 1. Distributions, differential operators and examples
- Introduction
- Topics to be treated in this book
- Partition of unity
- Test functions and distributions
- Convergence of test functions
- Space of test functions
- Distributions
- Differentiation of distributions
- The space C∞(Ω)
- Convergence properties of distributions
- Supports of distributions
- Distributions with compact support
- Convolution of a test function and a distribution
- Convolution of distributions
- Approximate identity
- Distributions and C∞-diffeomorphisms
- Tempered distributions and Fourier transforms
- Rapidly decreasing functions
- Tempered distributions
- Fourier transforms of tempered distributions
- Examples of Fourier transforms
- Convergence factors
- Partial Fourier transformation
- Chapter 2. Fundamental solutions
- Introduction and examples
- Hypo-elliptic operators
- Ordinary differential equations with constant coefficients
- Fundamental solutions of the Cauchy-Riemann operator
- Fundamental solutions of the Laplace equation in two dimensions
- Fundamental solutions of the heat equation
- Fundamental solutions of the Laplace operator in several space dimensions
- The free Schrodinger equation
- Chapter 3. Fundamental solutions of the wave operator
- Fundamental solutions of the wave operator in one space dimension
- Fundamental solutions of the wave equation in several space dimensions
- Fundamental solutions which are invariant under certain Lorentz transformations
- Explicit formulas for the fundamental solutions
- Chapter 4. Proofs of some main results
- Convolution products: formulation of some results
- Proofs
- Fourier transform and its inverse
- Riesz-Thorin interpolation
- Theorem of Malgrange and Ehrenpreis
- Sobolev theory
- Elliptic operators
- Sobolev spaces
- Quadratic forms and a compact embedding result
- Paley-Wiener theorems
- Multiplicative distributions
- The representation theorem for the dual of C0pXq
- Runge’s theorem
- Bibliography
- Index
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