This monograph guides the reader to the mathematical crossroads of heat equations and differential geometry via functional analysis. Following the recent trend towards constructive methods in the theory of partial differential equations, it makes extensive use of the ideas and techniques from the Weyl-Hörmander calculus of pseudo-differential operators to study heat Green operators through concrete calculations for the Dirichlet, Neumann, regular Robin and hypoelliptic Robin boundary conditions. Further, it provides detailed coverage of important examples and applications in elliptic and parabolic problems, illustrated with many figures and tables. A unified mathematical treatment for solving initial boundary value problems for the heat equation under general Robin boundary conditions is desirable, and leads to an extensive study of various aspects of elliptic and parabolic partial differential equations. The principal ideas are explicitly presented so that a broad spectrum of readers can easily understand the problem and the main results. The book will be of interest to readers looking for a functional analytic introduction to the meeting point of partial differential equations, differential geometry and probability.

Autorentext

Dr. Kazuaki Taira was awarded a Doctor of Science degree by the University of Tokyo (1976) and a Doctorat d'État degree by Université de Paris-Sud (Orsay) (1978), where he had studied on a French government scholarship (1976-1978). He was also a member of the Institute for Advanced Study (Princeton) (1980-1981), an associate professor at the University of Tsukuba (1981-1995), and a professor at Hiroshima University (1995-1998). In 1998, he accepted an offer from the University of Tsukuba to teach there again as a professor. He was also a part-time professor at Waseda University (2009-2017). His current research interests are in three interrelated subjects in analysis: semigroups, elliptic boundary value problems and Markov processes.