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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.
"The book is written carefully taking into account the interests of the reader, and it can be useful for scholars who work in the theory of partial differential equations and boundary value problems." (Vladimir Vasilyev, zbMATH 1555.35004, 2025)
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.
- Introduction and Main Results.- Part I Functional Analytic Approach to Heat Operators.- Preliminaries from Manifold Theory.- Finite Double-Norm and Trace-Class Operators on a Hilbert Space.- Hilbert-Schmidt and Trace-Class Operators on a Manifold.- Analytic Semigroups via Dunford Integrals.- Part II Pseudo-Differential Operators and Elliptic Boundary Value Problems.- Lp Theory of Pseudo-Differential Operators.- Elliptic Boundary Value Problems on a Manifold.- Lp Theory of Elliptic Boundary Value Problems.- Boutet de Monvel Calculus.- Pseudo-Differential Operator Approach to Agmon's Method.- Part III Analytic Semigroup Approach to Heat Operators.- Generation Theorem for Analytic Semigroups via Agmon's Method.- Hypoelliptic Robin Problems via Boutet de Monvel Calculus.- Distribution Kernel of Analytic Semigroups in the Hypoelliptic Case.- Part IV The Fundamental Solution for the Cauchy Problem.- The Cauchy Problem for the Heat Operator on Rn.- Fundamental Solution Operator for the Cauchy Problem on a Manifold.- Part V Symbolic Calculus for Dirichlet, Neumann and Regular Robin Problems.- Symbolic Calculus near the Boundary.- Analytic Version of Weyl Bases for the Heat Kernel on the Half Axis.- Symbolic Calculus for Boundary Value Problems on a Manifold.- Part VI Transport Equations and Trace Formulas on the Half Space.- Transport Equations via the Weyl-Hörmander Calculus.- Several Trace Formulas for Auxiliary Operators on the Half Space.- Part VII Heat Kernel Asymptotics for Dirichlet, Neumann and Regular Robin Problems.- Heat Kernel Asymptotics for the Dirichlet Problem.- Heat Kernel Asymptotics for the Neumann Problem.- Heat Kernel Asymptotics for the Regular Robin Problem.- Heat Kernel Asymptotics for the Generalized Regular Robin Problem.- Part VIII Heat Kernel Asymptotics for the Hypoelliptic Robin Problem.- Heat Kernel Asymptotics for the Hypoelliptic Robin Problem.- Examples in the Plane R2.- Concluding Remarks.
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