This careful and accessible text focuses on the relationship between two interrelated subjects in analysis: analytic semigroups and initial boundary value problems. This semigroup approach can be traced back to the pioneering work of Fujita and Kato on the Navier-Stokes equation. The author studies nonhomogeneous boundary value problems for second-order elliptic differential operators, in the framework of Sobolev spaces of Lp style, which include as particular cases the Dirichlet and Neumann problems, and proves that these boundary value problems provide an example of analytic semigroups in Lp.
Author(s): Kazuaki Taira
Series: London Mathematical Society Lecture Note Series
Publisher: CUP
Year: 1995
Language: English
Pages: 174
TABLE OF CONTENTS......Page 5
Preface......Page 9
Introduction and Results......Page 11
1.1 Generation Theorem for Analytic Semigroups......Page 18
1.2 Fractional Powers......Page 29
1.3 The Linear Cauchy Problem......Page 41
1.4 The Semilinear Cauchy Problem......Page 48
2.1 Holder Spaces and Sobolev Spaces......Page 56
2.2 Interpolation Theorems......Page 58
2.3 Imbeddings of the Spaces Hm*(Rn)......Page 84
2.4 Imbeddings of the Spaces Hm*{n)......Page 96
3.1 Generalized Sobolev Spaces and Besov Spaces......Page 103
3.2A Symbol Classes......Page 107
3.2B Phase Functions......Page 109
3.2C Oscillatory Integrals......Page 110
3.3 Pseudo-Differential Operators......Page 112
4.1 The Dirichlet Problem......Page 119
4.2 Formulation of a Boundary Value Problem......Page 121
4.3 Reduction to the Boundary......Page 125
4.4 Operator IT......Page 130
5.1 Regularity Theorem for Problem (*)......Page 132
5.2 Uniqueness Theorem for Problem (*)......Page 136
5.3 Existence Theorem for Problem (*)......Page 137
5.3A Proof of Theorem 5.7......Page 138
5.3B Proof of Proposition 5.10......Page 144
6.1 A Priori Estimates......Page 147
6.2 Generation of Analytic Semigroups......Page 152
7.1 Fractional Powers and Imbedding Theorems......Page 158
7.2B Proof of Theorem 4......Page 163
Appendix: The Maximum Principle......Page 167
References......Page 169
Index......Page 171
