Vector-valued Laplace Transforms and Cauchy Problems: Second Edition

553 Pages·2011·2.08 MB·English

by  Wolfgang Arendt

Monographs in Mathematics
Vol. 96

Managing Editors:
H. Amann
Universität Zürich, Switzerland
J.-P. Bourguignon
IHES, Bures-sur-Yvette, France
K. Grove
University of Maryland, College Park, USA
P.-L. Lions
Université de Paris-Dauphine, France

Associate Editors:
H. Araki, Kyoto University
F. Brezzi, Università di Pavia
K.C. Chang, Peking University
N. Hitchin, University of Warwick
H. Hofer, Courant Institute, New York
H. Knörrer, ETH Zürich
K. Masuda, University of Tokyo
D. Zagier, Max-Planck-Institut Bonn
Wolfgang Arendt
Charles J.K. Batty
Matthias Hieber
Frank Neubrander
Vector-valued
Laplace Transforms and
Cauchy Problems
Second Edition
Wolfgang Arendt Charles J.K. Batty
Angewandte Analysis St. John’s College
Universität Ulm Oxford OX1 3JP
89069 Ulm UK
Germany [email protected]
[email protected]

Matthias Hieber Frank Neubrander
Fachbereich Mathematik Department of Mathematics
TU Darmstadt Louisiana State University
Schlossgartenstr. 7 Baton Rouge, LA 70803
64289 Darmstadt USA
Germany [email protected]
[email protected]
2010 Mathematics Subject Classification: 35A22, 46F12, 35K25
ISBN 978-3-0348-0086-0 e-ISBN 978-3-0348-0087-7
DOI 10.1007/978-3-0348-0087-7
Library of Congress Control Number: 2011924209

© Springer Basel AG 2011
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Contents
Preface to the First Edition ix
Preface to the Second Edition xii
I Laplace Transforms and Well-Posedness of Cauchy Problems 1
1 The Laplace Integral 5
1.1 The Bochner Integral. . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.2 The Radon-Nikodym Property . . . . . . . . . . . . . . . . . . . . 15
1.3 Convolutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21
1.4 Existence of the Laplace Integral . . . . . . . . . . . . . . . . . . . 27
1.5 Analytic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6 Operational Properties . . . . . . . . . . . . . . . . . . . . . . . . . 36
1.7 Uniqueness, Approximation and Inversion . . . . . . . . . . . . . . 40
1.8 The Fourier Transform and Plancherel’s Theorem . . . . . . . . . . 44
1.9 The Riemann-Stieltjes Integral . . . . . . . . . . . . . . . . . . . . 48
1.10 Laplace-Stieltjes Integrals . . . . . . . . . . . . . . . . . . . . . . . 55
1.11 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2 The Laplace Transform 63
2.1 Riesz-Stieltjes Representation . . . . . . . . . . . . . . . . . . . . . 65
2.2 A Real Representation Theorem . . . . . . . . . . . . . . . . . . . 68
2.3 Real and Complex Inversion . . . . . . . . . . . . . . . . . . . . . . 73
2.4 Transforms of Exponentially Bounded Functions . . . . . . . . . . 77
2.5 Complex Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . 80
2.6 Laplace Transforms of Holomorphic Functions . . . . . . . . . . . . 84
2.7 Completely Monotonic Functions . . . . . . . . . . . . . . . . . . . 89
2.8 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
vi CONTENTS
3 Cauchy Problems 107
3.1 C -semigroups and Cauchy Problems . . . . . . . . . . . . . . . . . 108
0
3.2 Integrated Semigroups and Cauchy Problems . . . . . . . . . . . . 121
3.3 Real Characterization . . . . . . . . . . . . . . . . . . . . . . . . . 132
3.4 Dissipative Operators . . . . . . . . . . . . . . . . . . . . . . . . . 137
3.5 Hille-Yosida Operators . . . . . . . . . . . . . . . . . . . . . . . . . 141
3.6 Approximation of Semigroups . . . . . . . . . . . . . . . . . . . . . 145
3.7 Holomorphic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . 148
3.8 Fractional Powers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 162
3.9 Boundary Values of Holomorphic Semigroups . . . . . . . . . . . . 171
3.10 Intermediate Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . 184
3.11 Resolvent Positive Operators . . . . . . . . . . . . . . . . . . . . . 188
3.12 Complex Inversion and UMD-spaces . . . . . . . . . . . . . . . . . 197
3.13 Norm-continuous Semigroups and Hilbert Spaces . . . . . . . . . . 201
3.14 The Second Order Cauchy Problem. . . . . . . . . . . . . . . . . . 202
3.15 Sine Functions and Real Characterization . . . . . . . . . . . . . . 217
3.16 Square Root Reduction for Cosine Functions . . . . . . . . . . . . 222
3.17 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
II Tauberian Theorems and Cauchy Problems 239
4 Asymptotics of Laplace Transforms 243
4.1 Abelian Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 244
4.2 Real Tauberian Theorems . . . . . . . . . . . . . . . . . . . . . . . 247
4.3 Ergodic Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 261
4.4 The Contour Method . . . . . . . . . . . . . . . . . . . . . . . . . . 272
4.5 Almost Periodic Functions . . . . . . . . . . . . . . . . . . . . . . . 288
4.6 Countable Spectrum and Almost Periodicity. . . . . . . . . . . . . 295
4.7 Asymptotically Almost Periodic Functions . . . . . . . . . . . . . . 306
4.8 Carleman Spectrum and Fourier Transform . . . . . . . . . . . . . 318
4.9 Complex Tauberian Theorems: the Fourier Method . . . . . . . . . 325
4.10 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 329
5 Asymptotics of Solutions of Cauchy Problems 337
5.1 Growth Bounds and Spectral Bounds. . . . . . . . . . . . . . . . . 338
5.2 Semigroups on Hilbert Spaces . . . . . . . . . . . . . . . . . . . . . 351
5.3 Positive Semigroups . . . . . . . . . . . . . . . . . . . . . . . . . . 352
5.4 Splitting Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . 360
5.5 Countable Spectral Conditions . . . . . . . . . . . . . . . . . . . . 371
5.6 Solutions of Inhomogeneous Cauchy Problems . . . . . . . . . . . . 378
5.7 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 386
CONTENTS vii
III Applications and Examples 397
6 The Heat Equation 401
6.1 The Laplacian with Dirichlet Boundary Conditions . . . . . . . . . 401
6.2 Inhomogeneous Boundary Conditions. . . . . . . . . . . . . . . . . 408
6.3 Asymptotic Behaviour . . . . . . . . . . . . . . . . . . . . . . . . . 412
6.4 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 415
7 The Wave Equation 417
7.1 Perturbation of Selfadjoint Operators. . . . . . . . . . . . . . . . . 417
7.2 The Wave Equation in L2(Ω) . . . . . . . . . . . . . . . . . . . . . 423
7.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 427
8 Translation Invariant Operators on Lp(Rn) 429
8.1 Translation Invariant Operators and C -semigroups . . . . . . . . . 430
0
8.2 Fourier Multipliers . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
8.3 Lp-spectra and Integrated Semigroups . . . . . . . . . . . . . . . . 441
8.4 Systems of Differential Operators on Lp-spaces . . . . . . . . . . . 449
8.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 458
A Vector-valued Holomorphic Functions 461
B Closed Operators 467
C Ordered Banach Spaces 477
D Banach Spaces which Contain c 481
0
E Distributions and Fourier Multipliers 485
Indexes 493
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 493
Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 525
Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 533

Prefaces
Preface to the First Edition
Linear evolution equations in Banach spaces have seen important developments
in the last two decades. This is due to the many different applications in the
theory of partial differential equations, probability theory, mathematical physics,
and other areas, and also to the development of new techniques. One important
technique is given by the Laplace transform. It played an important role in the
earlydevelopmentofsemigrouptheory,ascanbeseeninthepioneeringmonograph
by Hille and Phillips [HP57]. But many new results and concepts have come from
Laplace transform techniques in the last 15 years. In contrast to the classical
theory, one particular feature of this method is that functions with values in a
Banach space have to be considered.
The aim of this bookis topresent the theory of linear evolution equations in
a systematic way by using the methods of vector-valued Laplace transforms.
It is simple to describe the basic idea relating these two subjects. Let A be a
closed linear operator on a Banach space X. The Cauchy problem defined by A is
the initial value problem
(cid:2)
u(cid:2)(t)= Au(t) (t≥0),
(CP)
u(0)= x,
where x∈X is a given initial value. If u is an exponentially bounded, continuous
function, then we may consider the Laplace transform
(cid:3)
∞
uˆ(λ)= e−λtu(t)dt
0
of u for large real λ. It turns out that u is a (mild) solution of (CP) if and only if
(λ−A)uˆ(λ)=x (λ large). (1)
Thus, if λ is in the resolvent set of A, then
uˆ(λ)=(λ−A)−1x. (2)