Partial differential equations and operators Fundamental solutions and semigroups Part I
Publisher: Bookboon
Published: 2013
ISBN-13: 9788740309997
Table of contents: title: Distributions, differential operators and examples; title: Test functions and distributions; title: Convergence of test functions; title: Space of test functions; title: Distributions; title: Differentiation of distributions; title: The space (Ω); title: Convergence properties of distributions; title: Supports of distributions; title: Distributions with compact support; title: Convolution of a test function and a distribution; title: Convolution of distributions; title: Approximate identity; title: Distributions and C8-diffeomorphisms; title: Tempered distributions and Fourier transforms; title: Rapidly decreasing functions; title: Tempered distributions; title: Fourier transforms of tempered distributions; title: Examples of Fourier transforms; title: Convergence factors; title: Partial Fourier transformation; title: Fundamental solutions; title: Introduction and examples; title: Hypo-elliptic operators; title: Ordinary differential equations with constant coefficients; title: Fundamental solutions of the Cauchy-Riemann operator; title: Fundamental solutions of the Laplace equation in two dimensions; title: Fundamental solutions of the heat equation; title: Fundamental solutions of the Laplace operator in several space dimensions; title: The free Schr¨odinger equation; title: Fundamental solutions of the wave operator; title: Fundamental solutions of the wave operator in one space dimension; title: Fundamental solutions of the wave equation in several space dimensions; title: Fundamental solutions which are invariant under certain Lorentz transformations; title: Explicit formulas for the fundamental solutions; title: Proofs of some main results; title: Convolution products: formulation of some results; title: Proofs; title: Fourier transform and its inverse; title: Riesz-Thorin interpolation; title: Theorem of Malgrange and Ehrenpreis; title: Sobolev theory; title: Elliptic operators; title: Sobolev spaces; title: Quadratic forms and a compact embedding result; title: Paley-Wiener theorems; title: Multiplicative distributions; title: The representation theorem for the dual of C0pXq; title: Runge’s theorem