Functional analytic methods for heat Green operators : heat kernel asymptotics via the Weyl-Hörmander calculus / Kazuaki Taira.

Format:

Book

Author/Creator:

Taira, Kazuaki, author.

Series:

Lecture notes in mathematics (Springer-Verlag) ; 2354.

Lecture notes in mathematics, 0075-8434 ; volume 2354

Language:

English

Subjects (All):

Functional analysis.

Physical Description:

xviii, 641 pages : illustrations ; 24 cm.

Place of Publication:

Cham, Switzerland : Springer, [2024]

Summary:

"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."-- Page 4 of cover.

Contents:

- 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.

Notes:

Includes bibliographical references (pages 623-632) and index.

ISBN:

9783031666117

3031666119

OCLC:

1440123880