Vector-Valved Laplace Transforms and Cauchy Problems

by Arendt, Wolfgang; Batty, Charles J. K.; Hieber, Matthias; Neubrander, Frank

• ISBN13: 9783034800860
• ISBN10: 303480086X
• Format: Hardcover
• Copyright: 2011-04-29
• Publisher: Birkhauser

Summary

This monograph gives a systematic account of the theory of vector-valued Laplace transforms, ranging from representation theory to Tauberian theorems. In parallel, the theory of linear Cauchy problems and semigroups of operators is developed completely in the spirit of Laplace transforms. Existence and uniqueness, regularity, approximation and above all asymptotic behaviour of solutions are studied. Diverse applications to partial differential equations are given. The book contains an introduction to the Bochner integral and several appendices on background material. It is addressed to students and researchers interested in evolution equations, Laplace and Fourier transforms, and functional analysis.

Table of Contents

Preface to the First Edition	p. ix
Preface to the Second Edition	p. xii
Laplace Transforms and Well-Posedness of Cauchy Problems	p. 1
The Laplace Integral	p. 5
The Bochner Integral	p. 6
The Radon-Nikodym Property	p. 15
Convolutions	p. 21
Existence of the Laplace Integral	p. 27
Analytic Behaviour	p. 32
Operational Properties	p. 36
Uniqueness, Approximation and Inversion	p. 40
The Fourier Transform and Plancherel's Theorem	p. 44
The Riemann-Stieltjes Integral	p. 48
Laplace-Stieltjes Integrals	p. 55
Notes	p. 59
The Laplace Transform	p. 63
Riesz-Stieltjes Representation	p. 65
A Real Representation Theorem	p. 68
Real and Complex Inversion	p. 73
Transforms of Exponentially Bounded Functions	p. 77
Complex Conditions	p. 80
Laplace Transforms of Holomorphic Functions	p. 84
Completely Monotonic Functions	p. 89
Notes	p. 100
Cauchy Problems	p. 107
C0-semigroups and Cauchy Problems	p. 108
Integrated Semigroups and Cauchy Problems	p. 121
Real Characterization	p. 132
Dissipative Operators	p. 137
Hille-Yosida Operators	p. 141
Approximation of Semigroups	p. 145
Holomorphic Semigroups	p. 148
Fractional Powers	p. 162
Boundary Values of Holomorphic Semigroups	p. 171
Intermediate Spaces	p. 184
Resolvent Positive Operators	p. 188
Complex Inversion and UMD-spaces	p. 197
Norm-continuous Semigroups and Hilbert Spaces	p. 201
The Second Order Cauchy Problem	p. 202
Sine Functions and Real Characterization	p. 217
Square Root Reduction for Cosine Functions	p. 222
Notes	p. 230
Tauberian Theorems and Cauchy Problems	p. 239
Asymptotics of Laplace Transforms	p. 243
Abelian Theorems	p. 244
Real Tauberian Theorems	p. 247
Ergodic Semigroups	p. 261
The Contour Method	p. 272
Almost Periodic Functions	p. 288
Countable Spectrum and Almost Periodicity	p. 295
Asymptotically Almost Periodic Functions	p. 306
Carleman Spectrum and Fourier Transform	p. 318
Complex Tauberian Theorems: the Fourier Method	p. 325
Notes	p. 329
Asymptotics of Solutions of Cauchy Problems	p. 337
Growth Bounds and Spectral Bounds	p. 338
Semigroups on Hilbert Spaces	p. 351
Positive Semigroups	p. 352
Splitting Theorems	p. 360
Countable Spectral Conditions	p. 371
Solutions of Inhomogeneous Cauchy Problems	p. 378
Notes	p. 386
Applications and Examples	p. 397
The Heat Equation	p. 401
The Laplacian with Dirichlet Boundary Conditions	p. 401
Inhomogeneous Boundary Conditions	p. 408
Asymptotic Behaviour	p. 412
Notes	p. 415
The Wave Equation	p. 417
Perturbation of Selfadjoint Operators	p. 417
The Wave Equation in L2(¿)	p. 423
Notes	p. 427
Translation Invariant Operators on Lp(Rn)	p. 429
Translation Invariant Operators and C0-semigroups	p. 430
Fourier Multipliers	p. 435
Lp-spectra and Integrated Semigroups	p. 441
Systems of Differential Operators on Lp-spaces	p. 449
Notes	p. 458
Vector-valued Holomorphic Functions	p. 461
Closed Operators	p. 467
Ordered Banach Spaces	p. 477
Banach Spaces which Contain c0	p. 481
Distributions and Fourier Multipliers	p. 485
Indexes	p. 493
Bibliography	p. 493
Notation	p. 525
Index	p. 533
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