Van Casteren J.A. Partial differential equations and operators. Fundamental solutions and semigroups. Part I

3rd. ed. — Bookboon.com, 2018. — 172 p.The contents of this book came into existence during the time the author has been teaching a course on partial differential equations and on operator theory for about twenty years. Let us discuss the contents somewhat more closely. Basically, the book contains topics from distribution theory, from operator theory, and from operator semigroups. These lecture notes follow to some extent the book by Rudin. The subject of distribution theory is covered to a large extent: distributions and their derivatives, compactly supported distributions, tempered distributions and their Fourier transforms, convolution products of distributions, existence of fundamental solutions of partial differential operators with constant coefficients, several concrete examples, multiplicative distributions. Most of these topics can be found in Chapter 1. The Chapters 2 and 3 contain several concrete examples of fundamental solution (for the heat equation, free Schrodinger equation, for the Laplace equation, and for the wave equation. Some theory on Banach algebras and (unbounded) operators in Hilbert space can be found in Chapter 5. Chapter 6 is devoted to some aspects of operator semigroup theory. The chapter on operators in Hilbert space contains some basic results on Banach algebras, like spectral theory, symbolic calculus, and Gelfand transforms. It also contains a discussion on commutative C˚ -algebras, including the Gelfand-Naimark theorem, symbolic calculus, taking square roots. Unbounded closed, and closable operators are treated as well. Again the spectral mapping theorem for normal operators is included, and also the polar decomposition. The chapter on operator semigroups discusses some basic generation results, like the Hille-Yosida theorem, and the Lumer-Phillips theorem. The maximum principle, and its connection with dissipative operators is explained. It also contains a result which shows the close connection with initial value problems. In addition, relations with Markov processes are described. In Chapter 6 some attention is paid to Feynman-Kac semigroups, and it concludes with a discussion on the KMS-function. In Chapter 1 the fundamental ideas for the theoretical concepts of distribution theory, or generalize function theory, are explained. Many (continuity) results about distributions are proved in Chapter 4. Chapter 7 contains a discussion on (bounded) analytic semigroups. It includes a discussion on the Crank-Nicolson iteration scheme. The final Chapter 8 contains some results on Functional analysis, which are used in the other chapters of the book. Most, if not all, functional analytic aspects of the course are explained. The final chapter serves as sort of a reference. In the present version several corrections were made, and some new text is included.
Readership. From the description of the contents it is clear that the text is designed for students at the graduate or master level. The author believes that also Ph.D. students, and even researchers, might benefit from these notes. The reader is introduced to the following topics: generalized function or distribution theory, operators in Hilbert space (including rudiments of C˚ -algebras) and their spectral decompositions, and operator semigroup theory.Distributions, differential operators and examples.
Fundamental solutions.
Fundamental solutions of the wave operator.
Proofs of some main results.