Applied Semigroups and Evolution Equations

✍ Scribed by Aldo Belleni-Morante

• Publisher: Oxford University Press, USA
• Year: 1979
• Tongue: English
• Leaves: 406
• Series: Oxford Mathematical Monographs
• Category: Library

✦ Synopsis

This book is mainly intended for applied mathematicians, physicists, and engineers and, as such, it gives a self contained introduction to the theory of semigroups and of linear and semilinear evolution-equations in Banach spaces, with particular emphasis on applications to concrete problems from mathematical physics.

Since the only prerequisite is a good knowledge of classical differential and integral calculus, the first three chapters give a 'compact picture' of Banach and Hilbert spaces and introduce the basic notions of abstract differential and integral calculus. Chapters 4 and 5 deal with semigroups and with their applications to linear and semilinear evolution equations. In Chapter 6, a detailed discussion is presented on how a problem of evolution in a given Banach space can be approximated by means of a sequence of problems in the same space or in different spaces. The relationships between the spectral properties of generators and those of semigroups are discussed in Chapter 7. Definitions and theorems of Chapters 1-7 are always supplemented with several examples completely worked out. Finally, each of the final six chapters is devoted to a complete study of a problem from applied mathematics, by using the techniques developed in the previous chapters.

This book is based on lectures given by the author to final-year undergraduates and to first-year graduates of the Mathematical Schools of Bari University and of Florence University, and on seminars given in the mathematical department of Oxford University.

Readers who want to arrive quickly at 'where the action is' may skip (in a first reading) Sections 1.4, 1.5 on Sobolev spaces (and all the Examples that deal with these spaces), Chapters 6 and 7, and some of the proofs in Chapters 2-5.

✦ Table of Contents

Cover

OXFORD MATHEMATICAL MONOGRAPHS

APPLIED SEMIGROUPS AND EVOLUTION EQUATIONS

Copyright
Oxford University Press 1979
ISBN 0 19 853529 5

Dedicated To Sara

PREFACE

CONTENTS

INTRODUCTION

1 BANACH AND HILBERT SPACES

 1.1. BANACH AND HlILBERT SPACES

 1.2. EXAMPLES OF BANACH AND HILBERT SPACES

 1.3. GENERALIZED DERIVATIVES

 1.4. SOBOLEV SPACES OF INTEGER ORDER

 1.5. SOBOLEV SPACES OF FRACTIONAL ORDER

 EXERCISES

2 OPERATORS IN BANACH SPACES

 2.1. NOTATION AND BASIC DEFINITIONS

 2.2. BOUNDED LINEAR OPERATORS

 2.3. EXAMPLES OF LINEAR BOUNDED OPERATORS

 2.4. LIPSCHITZ OPERATORS

 2.5. CLOSED OPERATORS

 2.6. SELF-ADJOINT OPERATORS

 2.7. SPECTRAL PROPERTIES: BASIC DEFINITIONS

 2.8. SPECTRAL PROPERTIES : EXAMPLES

 EXERCISES

3 ANALYSIS IN BANACH SPACES

 3.1. STRONG CONTINUITY

 3.2. STRONG DERIVATIVE

 3.3. STRONG RIEMANN INTEGRAL

 3.4. THE DIFFERENTIAL EQUATION du/dt = F(u)

 3.5. HOLOMORPHIC FUNCTIONS

 EXERCISES

4 SEMIGROUPS

 4.1. LINEAR INITIAL-VALUE PROBLEMS

 4.2. THE CASE A in B(X )

 4.3. THE CASE A in C(X )

 4.4. THE CASE A in G(1, 0; X): TWO PRELIMINARY LEMMAS

 4.5. THE SEMI GROUP GENERATED BY A in  G(1, O ; X)

 4.6. THE CASES A in S(M,0 ;X) ,Bn(M, \beta; X) ,S' (M, S; X)

 4.7. THE HOMOGENEOUS AND THE NON-HOMOGENEOUS INITIAL-VALUE PROBLEMS

 EXERCISES

5 PERTURBATION THEOREMS

 5.1. INTRODUCTION

 5.2. BOUNDED PERTURBATIONS

 5.3. THE CASES B = B(t) in B(X) AND \beta RELATIVELY BOUNDED

 5.4. THE SEMILINEAR CASE

 5.5. GLOBAL SOLUTION OF THE SEMI LI NEAR PROBLEM (5.31)

 EXERCISES

6 SEQUENCES OF SEMIGROUPS

 6.1. SEQUENCES OF SEMI GROUPS exp (tA j)

 6.2. SEQUENCES OF BANACH SPACES

 6.3. SEQUENCES OF SEMI GROUPS exp (tA.) E

 EXERCISES

7 SPECTRAL REPRESENTATION OF CLOSED OPERATORS AND OF SEMIGROUPS

 7.1. INTRODUCTION

 7.2. PROJECTIONS

 7.3. ISOLATED POINTS OF THE SPECTRUM OF A in G(X)

 7.4. LAURENT EXPANSION OF R (z ,A)

 7.5. ISOLATED E I GENVALUES

 7.6. SPECTRAL REPRESENTATION OF A AND OF exp(tA)

 EXERCISES

8 HEAT CONDUCTION IN RIGID BODIES AND SIMILAR PROBLEMS

 8.1. INTRODUCTION

 8.2. A LINEAR HEAT-CONDUCTION PROBLEM IN L^2(a,b)

 8.3. A SEMILINEAR HEAT-CONDUCTION PROBLEM

 8.4. POSITIVE SOLUTIONS

 EXERCISES

9 NEUTRON TRANSPORT

 9.1. 'INTRODUCTION

 9.2. LINEAR NEUTRON TRANSPORT IN L^2((-a,a)x(-1,1))

 9.3. SPECTRAL PROPERTIES OF THE TRANSPORT OPERATOR A 2

 9.4. A SEMILINEAR NEUTRON TRANSPORT PROBLEM

 EXERCISES

10 A SEMILINEAR PROBLEM FROM KINETIC THEORY OF VEHICULAR TRAFFIC

 10.1. INTRODUCTION

 10.2. PRELIMINARY LEMMAS

 10.3. THE OPERATORS F. K1 AND K2

 10.4. THE OPERATORS J AND K3

 10.5. GLOBAL SOLUTION OF THE ABSTRACT PROBLEM (10.11)

 EXERCISES

11 THE TELEGRAPHIC EQUATION AND THE WAVE EQUATION

 11.1. INTRODUCTION

 11.2. PRELIMINARY LEMMAS

 11.3. THE ABSTRACT VERSION OF THE TELEGRAPHIC SYSTEM (11.4)

 11.4. THE TELEGRAPHIC EQUATION AND THE WAVE EQUATION

 EXERCISES

12 A PROBLEM FROM QUANTUM MECHANICS

 12.1. INTRODUCTION

 12.2. SPECTRAL PROPERTIES OF iA

 12.3. BOUNDED PERTURBATIONS

 EXERCISES

13 A PROBLEM FROM STOCHASTIC POPULATION THEORY

 13.1. INTRODUCTION

 13.2. THE ABSTRACT PROBLEM

 13.3. PRELIMINARY LEMMAS

 13.4. STRICT SOLUTION OF THE APPROXIMATING PROBLEM (13.7)

 13.5. A PROPERTY OF THE STRICT SOLUTION OF THE APPROXIMATING PROBLEM

 13.6. STRICT SOLUTION OF PROBLEM (13.6)

 13.7. THE EQUATION FOR THE FIRST MOMENT (n) (t) OF THE BACTERIA POPULATION

 EXERCISES

BIBLIOGRAPHY

SUBJECT INDEX

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