Existence Families, Functional Calculi and Evolution Equations
This book presents an operator-theoretic approach to ill-posed evolution equations. It presents the basic theory, and the more surprising examples, of generalizations of strongly continuous semigroups known as 'existent families' and 'regularized semigrou
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1570
Lecture Notes in Mathematics Editors: A. Dold, Heidelberg B. Eckmann, Zurich F. Takens, Groningen
1570
Ralph deLaubenfels
Existence Families, Functional Calculi and Evolution Equations
Springer-Verlag Berlin Heidelberg New York London Paris Tokyo Hong Kong Barcelona Budapest
Author Ralph deLaubenfels Mathematics Department Ohio University 315 B Morton Hall Athens, Ohio 45701-2979, USA
Mathematics Subject Classification (1991): 47D05, 47D 10,47A60, 34G 10, 35G 10, 47A1O, 47A12, 47F05
ISBN' 3-540-57703-3 Springer-Verlag Berlin Heidelberg New York ISBN 0-387-57703-3 Springer-Verlag New York Berlin Heidelberg Library of Congress Cataloging-in-Publication Data. DeLaubenfels, Ralph, 1951- . Existence families, functional calculi, and evolution equations / Ralph deLaubenfels. p. ern, - (Lecture notes in mathematics; 1570) Includes bibliographical references and index. ISBN 0-387-57703-3 1.Evolution equations. 2. Linear operators. 3. Functional analysis. I. Title. II. Series: Lecture notes in mathematics (Springer-Verlag); 1570. QA3.L28 no. 1570 [QA377] 510 s-dc20 [515'.353] 93-47576 This work is subject to copyright. All rights are reserved, whether the whole or part ofthe material is concerned, specifically the rights of translation, reprinting, re-use of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9, 1965, in its current version, and permission for use must always be obtained from Springer-Verlag. Violations are liable for prosecution under the German Copyright Law. © Springer-Verlag Berlin Heidelberg 1994 Printed in Germany SPIN: 10078796
46/3140-543210 - Printed on acid-free paper
To Karen, David, Michael and Evan
TABLE OF CONTENTS
O.
Introduction
I.
Intuition and Elementary Examples
1
II.
Existence Families
7
III.
Regularized Semigroups
13
IV.
The Solution Space of an Operator and Automatic Well-Posedness
24
V.
Exponentially Bounded (Banach) Solution Space
38
VI.
Well-posedness on a Larger Space; Generalized Solutions
55
IX
VII. Entire Vectors and Entire Existence Families
60
VIII. Reversibility of Parabolic Problems
66
IX.
The Cauchy Problem for the Laplace Equation
71
X.
Boundary Values of Holomorphic Semigroups
73
XI.
The Schr6dinger Equation
76
XII. Functional Calculus for Commuting Generators of Bounded Strongly Continuous Groups
79
XIII. Petrovsky Correct Matrices of Generators of Bounded Strongly Continuous Groups
86
XIV. Arbitrary Matrices of Generators of Bounded Strongly Continuous Groups
92
XV. More Examples of Regularized Semigroups
94
XVI. Existence and Uniqueness Families
97
VII
XVII.
C-resolvents and Hille-Yosida Type Theorems
104
XVIII.
Relationship to Integrated Semigroups
110
XIX.
Perturbations
113
XX.
Type of an Operator
125
XXI.
Holomorphic C -existence Families
128
XXII.
Unbounded Holomorphic Functional Calculus for
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