2001 • 526 Pages • 13.5 MB • English

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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 Universite de Paris-Dauphine, France Associate Editors: H. Araki, Kyoto University F. Brezzi, Universiat di Pavia K.C. Chang, Peking University N. Hitchin, University ofWarwick H. Hofer, Courant Institute, New York H. Knörrer, ETH Zürich K. Masuda, University of Tokyo D. Zagier, Max-Planck-Institut Bonn

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Wolfgang Arendt Charles J.K. Batty Matthias Hieber Frank Neubrander Vector-valued Laplace Transforms and Cauchy Problems Springer Basel AG

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Authors: Wo1fgang Arendt Charles J.K. Batty Angewandte Analysis St. John's College Universitiit Ulm Oxford OX1 3JP 89069Ulm UK Germany e-mail: [email protected] e-mail: [email protected] Matthias Hieber Frank Neubrander Fachbereich Mathematik Departrnent of Mathematics TU Darmstadt Louisiana State University Schlossgartenstr. 7 Baton Rouge, LA 70803 64289 Darmstadt USA Germany e-mail: [email protected] e-mail: [email protected] 2000 Mathematics Subject Classification 35A22, 46F12, 35K25 A CIP catalogue record for this book is available from the Library of Congress, Washington D.C., USA Vector-valued Laplace transforms and Cauchy problems 1W olfgang Arendt ... [et al.]. p. cm. - (Monographs in mathematics ; voi. 96) lncludes bibliographical references and index. ISBN 978-3-0348-5077-3 ISBN 978-3-0348-5075-9 (eBook) DOI 10.1007/978-3-0348-5075-9 1. Laplace transformation. 2. Cauchy problem. 1. Wolfgang, Arendt, 1950- II. Monographs in mathematics ; v. 96. QA432.V43 2001 515'.723- dc21 Deutsche Bib1iothek Cataloging-in-Publication Data Vector valued Laplace transforms and Cauchy prob1ems 1 Wo1fgang Arendt ... (Monographs in mathematics ; Voi. 96) ISBN 978-3-0348-5077-3 This work is subject to copyright. Ali rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in other ways, and storage in data banks. For any kind of use the permis- sion of the copyright owner must be obtained. © 2001 Springer Basel AG Originally published by Birkhauser Verlag, Basel- Boston- Berlin in 2001 Softcover reprint of the hardcover 1s t edition 2001 Member of the BertelsmannSpringer Publishing Group Printed on acid-free paper produced from chlorine-free pulp. TCF oo ISBN 978-3-0348-5077-3 987654321 www.birkhauser-science.com

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Contents Preface A Laplace Transforms and Well-Posedness of Cauchy Problems 1 The Laplace Integral 1.1 The Bochner Integral . . . . . . 6 1.2 The Radon-Nikodym Property 15 1.3 Convolutions ......... . 21 1.4 Existence of the Laplace Integral 28 1.5 Analytic Behaviour ....... . 33 1.6 Operational Properties . . . . . . 36 1.7 Uniqueness, ApproximationandInversion 40 1.8 The Fourier Transform and Plancherel's Theorem . 45 1.9 The Riemann-Stieltjes Integral 49 1.10 Laplace-Stieltjes Integrals 56 1.11 Notes ............ . 61 2 The Laplace Transform 2.1 Riesz-Stieltjes Representation 67 2.2 AReal Representation Theorem 70 2.3 Real and Camplex Inversion ... 75 2.4 Transforms of Exponentially Bounded Functions 79 2.5 Camplex Conditions .............. . 83 2.6 Laplace Transforms of Holamorphie Functions . 87 2. 7 Completely Monotonic Functions 92 2.8 Notes ...................... . 103 3 Cauchy Problems 3.1 Co-semigroups and Cauchy Problems . 110 3.2 Integrated Semigroups and Cauchy Problems 123 3.3 Real Characterization 135 3.4 Dissipative Operators . . . . 140 3.5 Hille-Yosida Operators .... 144 3.6 Approximation of Semigroups 149

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vi Contents 3. 7 Holomorphic Semigroups . . . . . . . . . . . . 151 3.8 Fractional Powers ............... . 166 3.9 Boundary Values of Holomorphic Semigroups 174 3.10 Intermediate Spaces ........ . 187 3.11 Resolvent Positive Operators ........ . 191 3.12 Complex Inversion and UMD-spaces .... . 200 3.13 Norm-continuous Semigroups and Hilbert Spaces 204 3.14 The Second Order Cauchy Problem ..... . 206 3.15 Sine Functions and Real Characterization .. 221 3.16 Square Root Reduction for Cosine Functions 226 3.17 Notes ..................... . 234 B Tauberian Theorems and Cauchy Problems 4 Asymptotics of Laplace Transforms 4.1 Abelian Theorems .... 246 4.2 Real Tauberian Theorems 249 4.3 Ergodie Semigroups . . . 263 4.4 The Gontour Method . . . 275 4.5 Almost Periodic Functions . 285 4.6 Countahle Spectrum and Almost Periodicity . 292 4. 7 Asymptotically Almost Periodic Functions . . 304 4.8 Carleman Spectrum and Fourier Transform 316 4.9 Complex Tauberian Theorems: the Fourier Method . 323 4.10 Notes ......................... . 328 5 Asymptotics of Solutions of Cauchy Problems 5.1 Growth Bounds and Spectral Bounds. 334 5.2 Semigroups on Hilbert Spaces 347 5.3 Positive Semigroups . . . . . . 349 5.4 Splitting Theorems . . . . . . . 356 5.5 Countahle Spectral Conditions 367 5.6 Solutions of Inhomogeneous Cauchy Problems . 374 5.7 Notes ...................... . 383 C Applications and Examples 6 The Heat Equation 6.1 The Laplacian with Dirichlet Boundary Conditions 395 6.2 Inhomogeneous Boundary Conditions. 402 6.3 Asymptotic Behaviour 407 6.4 Notes ................. . 410

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Contents vii 7 The Wave Equation 7.1 Perturbation of Seifadjoint Operators. 411 7.2 The Wave Equation in L2 (!1) .... . 417 7.3 Notes ................. . 421 8 Translation Invariant Operators on LP (lRn) 8.1 Translation Invariant Operators and C0-semigroups . 424 8.2 Fourier Multipliers . . . . . . . . . . . . . . . . 429 8.3 LP-spectra and Integrated Semigroups ..... 436 8.4 Systems of Differential Operators on LP-spaces 443 8.5 Notes ...................... . 453 Appendices A Vector-valued Holomorphic Functions 455 B Closed Operators . . . . 461 C Ordered Banach Spaces 471 D Banach Spaces which Contain c0 475 E Distributions and Fourier Multipliers 479 Bibliography 487 Notation. 511 Index .. 517

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Preface 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 early development of semigroup theory, as can be seen in the pioneering monograph by Rille 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 book is to present 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 problern defined by A is the initial value problern (CP) {u'(t) = Au(t) (t 2 0), u(O) = x, where x E X is a given initial value. If u is an exponentially bounded, continuous function, then we may consider the Laplace transform 00 u(>.) = 1 e-)..tu(t) dt of u for large real>.. It turnsout that u is a (mild) solution of (CP) if and only if (>.- A)u(>.) = x (>. large). (1) Thus, if >. is in the resolvent set of A, then (2) Now it is a typical feature of concrete evolution equations that no explicit infor- mation on the solution is known and only in exceptional cases can the solution be

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X Preface given by a formula. On the other hand, in many cases much can be said about the resolvent of the given operator. The fact that the Laplace transform allows us to reduce the Cauchy problern (C P) to the characteristic equation (1) explains its usefulness. The Laplace transform is the link between solutions and resolvents, between Cauchy problems and spectral properties of operators. There are two important themes in the theory of Laplace transforms. The first concerns representation theorems; i.e., results which give criteria to decide whether a given function is a Laplace transform. Clearly, in view of (2), such results, applied to the resolvent of an operator, give information on the solvability of the Cauchy problem. The other important subject is asymptotic behaviour, where the most chal- lenging and delicate results are Tauberian theorems which allow one to deduce asymptotic properties of a function from properties of its transform. Since in the case of solutions of (C P) the transform is given by the resolvent, such results may allow one to deduce results of asymptotic behaviour from spectral properties of A. These two themes describe the essence of this book, which is divided into three parts. In the first, representation theorems for Laplace transforms are given, and corresponding to this, well-posedness of the Cauchy problern is studied. The second is a systematic study of asymptotic behaviour of Laplace transforms first of arbitrary functions, and then of solutions of (C P). The last part contains appli- cations and illustrative examples. Each part is preceded by a detailed introduction where we describe the interplay between the diverse subjects and explain how the sections are related. We have assumed that the reader is already familiar with the basic topics of functional analysis and the theory of bounded linear operators, Lebesgue inte- gration and functions of a complex variable. We require some standard facts from Fourier analysis and slightly more advanced areas of functional analysis for which we give references in the text. There are also four appendices (A, B, C and E) which collect together background material on other standard topics for use in various places in the book, while Appendix D gives a proof of a technical result in the geometry of Banach spaces which is needed in Section 4.6. Finally, a few words should be said about the realization of the book. The collaboration of the authors is based on two research activities: the common work of W. Arendt, M. Hieber and F. Neubrander on integrated semigroups and the work of W. Arendt and C. Batty on asymptotic behaviour of semigroups over many years. Laplace transform methods are common to both. The actual contributions are as follows. Part I: All four authors wrote this part. Part II was written by W. Arendt and C. Batty. Part III was written by W. Arendt (Chapters 6 and 7) and M. Hieber (Chap- ter 8). C. Batty undertook the coordination needed to make the material into a consistent book.

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