Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential Equations

503 Pages·2020·6.329 MB·English

by  Luca Lorenzi, Abdelaziz Rhandi

Semigroups of Bounded
Operators and Second-Order
Elliptic and Parabolic Partial
Differential Equations
Monographs and Research Notes in Mathematics
Series Editors:
John A. Burns, Thomas J. Tucker, Miklos Bona, Michael Ruzhansky
About the Series
This series is designed to capture new developments and summarize what is known over the entire
field of mathematics, both pure and applied. It will include a broad range of monographs and
research notes on current and developing topics that will appeal to academics, graduate students,
and practitioners. Interdisciplinary books appealing not only to the mathematical community, but
also to engineers, physicists, and computer scientists are encouraged.
This series will maintain the highest editorial standards, publishing well-developed monographs as
well as research notes on new topics that are final, but not yet refined into a formal monograph. The
notes are meant to be a rapid means of publication for current material where the style of exposition
reflects a developing topic.
Summable Spaces and Their Duals, Matrix Transformations and Geometric Properties
Feyzi Basar, Hemen Dutta
Spectral Geometry of Partial Differential Operators (Open Access)
Michael Ruzhansky, Makhmud Sadybekov, Durvudkhan Suragan
Linear Groups: The Accent on Infinite Dimensionality
Martyn Russel Dixon, Leonard A. Kurdachenko, Igor Yakov Subbotin
Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, Volume I
Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim
Morrey Spaces: Introduction and Applications to Integral Operators and PDE’s, Volume II
Yoshihiro Sawano, Giuseppe Di Fazio, Denny Ivanal Hakim
Tools for Infinite Dimensional Analysis
Jeremy J. Becnel
Semigroups of Bounded Operators and Second-Order Elliptic and Parabolic Partial Differential
Equations
Luca Lorenzi, Abdelaziz Rhandi
For more information about this series please visit: https://www.crcpress.com/Chapman--HallCRC-
Monographs-and-Research-Notes-in-Mathematics/book-series/CRCMONRESNOT
Semigroups of Bounded
Operators and Second-Order
Elliptic and Parabolic Partial
Differential Equations
Luca Lorenzi
University of Parma, Italy
Abdelaziz Rhandi
University of Salerno, Italy
First edition published 2021
by CRC Press
6000 Broken Sound Parkway NW, Suite 300, Boca Raton, FL 33487-2742
and by CRC Press
2 Park Square, Milton Park, Abingdon, Oxon, OX14 4RN
© 2021 Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, LLC
Reasonable efforts have been made to publish reliable data and information, but the author and
publisher cannot assume responsibility for the validity of all materials or the consequences of
their use. The authors and publishers have attempted to trace the copyright holders of all mate-
rial reproduced in this publication and apologize to copyright holders if permission to publish
in this form has not been obtained. If any copyright material has not been acknowledged please
write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, repro-
duced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now
known or hereafter invented, including photocopying, microfilming, and recording, or in any
information storage or retrieval system, without written permission from the publishers.
For permission to photocopy or use material electronically from this work, access www.
copyright.com or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood Drive,
Danvers, MA 01923, 978-750-8400. For works that are not available on CCC please contact
[email protected]
Trademark notice: Product or corporate names may be trademarks or registered trademarks and
are used only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
ISBN: 9780367206291(hbk)
ISBN: 9780429262593 (ebk)
v
To Adriana, Alfredo, Asma, Ursula and Rainer.

Contents
Symbol Description xi
Preface xiii
Introduction xv
1 Function Spaces 1
1.1 Spaces of (H¨older) Continuous Functions . . . . . . . . . . . . . . . . . . . 1
1.1.1 Functions defined on the boundary of a smooth open set . . . . . . . 7
1.2 Anisotropic and Parabolic Spaces of H¨older Continuous Functions . . . . . 8
1.2.1 Anisotropic spaces of functions defined on the boundary of a set . . 14
1.3 Lp- and Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15
1.4 Besov Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34
I Semigroups of Bounded Operators 35
2 Strongly Continuous Semigroups 37
2.1 Definitions and Basic Properties . . . . . . . . . . . . . . . . . . . . . . . . 38
2.2 The Infinitesimal Generator . . . . . . . . . . . . . . . . . . . . . . . . . . 40
2.3 The Hille-Yosida, Lumer-Phillips and Trotter-Kato Theorems . . . . . . . . 44
2.4 Nonhomogeneous Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . 50
2.5 Notes and Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53
2.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
3 Analytic Semigroups 55
3.1 Prelude . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56
3.2 Sectorial Operators and Analytic Semigroups . . . . . . . . . . . . . . . . . 57
3.3 Interpolation Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
3.4 Nonhomogeneous Cauchy Problems . . . . . . . . . . . . . . . . . . . . . . 70
3.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 79
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
II Parabolic Equations 83
4 Elliptic and Parabolic Maximum Principles 85
4.1 The Parabolic Maximum Principles . . . . . . . . . . . . . . . . . . . . . . 85
4.1.1 Parabolic weak maximum principle . . . . . . . . . . . . . . . . . . . 85
4.1.2 The strong maximum principle . . . . . . . . . . . . . . . . . . . . . 92
4.2 Elliptic Maximum Principles . . . . . . . . . . . . . . . . . . . . . . . . . . 97
4.3 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
vii
viii Contents
5 Prelude to Parabolic Equations: The Heat Equation and the Gauss-
Weierstrass Semigroup in C (Rd) 107
b
5.1 The Homogeneous Heat Equation in Rd. Classical Solutions: Existence and
Uniqueness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108
5.2 The Gauss-Weierstrass Semigroup . . . . . . . . . . . . . . . . . . . . . . . 111
5.2.1 Estimates of the spatial derivatives of T(t)f . . . . . . . . . . . . . . 114
5.3 Two Equivalent Characterizations of H¨older Spaces . . . . . . . . . . . . . 120
5.4 Optimal Schauder Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . 125
5.5 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
5.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136
6 Parabolic Equations in Rd 137
6.1 The Continuity Method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 138
6.2 A priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
6.2.1 Solving problem (6.0.1) . . . . . . . . . . . . . . . . . . . . . . . . . 145
6.2.2 Interior Schauder estimates for solutions to parabolic equations in
domains: Part I . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
6.3 More on the Cauchy Problem (6.0.1) . . . . . . . . . . . . . . . . . . . . . 157
6.4 The Semigroup Associated with the Operator A . . . . . . . . . . . . . . . 163
6.4.1 Interior Schauder estimates for solutions to parabolic equations in
domains: Part II . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
6.5 Higher-Order Regularity Results . . . . . . . . . . . . . . . . . . . . . . . . 171
6.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
6.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 174
7 Parabolic Equations in Rd with Dirichlet Boundary Conditions 175
+
7.1 Technical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 176
7.2 An Auxiliary Boundary Value Problem . . . . . . . . . . . . . . . . . . . . 184
7.3 Proof of Theorem 7.0.2 and a Corollary . . . . . . . . . . . . . . . . . . . . 190
7.4 More on the Cauchy Problem (7.0.1) . . . . . . . . . . . . . . . . . . . . . 192
7.5 The Associated Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 193
7.6 Notes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
7.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 197
8 Parabolic Equations in Rd with More General Boundary Conditions 199
+
8.1 A Priori Estimates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 200
8.2 Proof of Theorem 8.0.2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 209
8.3 InteriorSchauderEstimatesforSolutionstoParabolicEquationsinDomains:
Part III . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 219
8.4 More on the Cauchy Problem (8.0.1) . . . . . . . . . . . . . . . . . . . . . 224
8.5 The Associated Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 225
8.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 228
9 Parabolic Equations in Bounded Smooth Domains Ω 229
9.1 Optimal Schauder Estimates for Solutions to Problems (9.0.1) and (9.0.2) . 230
9.2 InteriorSchauderEstimatesforSolutionstoParabolicequationsinDomains:
Part IV . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 240
9.3 More on the Cauchy Problems (9.0.1) and (9.0.2) . . . . . . . . . . . . . . 243
9.4 The Associated Semigroup . . . . . . . . . . . . . . . . . . . . . . . . . . . 245
9.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
Contents ix
III Elliptic Equations 247
10 Elliptic Equations in Rd 249
10.1 Solutions in H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 251
10.1.1 The Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . 251
10.1.2 More general elliptic operators . . . . . . . . . . . . . . . . . . . . . 252
10.1.3 Further regularity results and interior estimates . . . . . . . . . . . . 254
10.2 Solutions in Lp(Rd;C) (p∈(1,∞)) . . . . . . . . . . . . . . . . . . . . . . 256
10.2.1 The Calder´on-Zygmund inequality . . . . . . . . . . . . . . . . . . . 256
10.2.2 The Laplace equation . . . . . . . . . . . . . . . . . . . . . . . . . . 266
10.2.3 More general elliptic operators . . . . . . . . . . . . . . . . . . . . . 269
10.2.4 Further regularity results and interior Lp-estimates . . . . . . . . . . 279
10.3 Solutions in L∞(Rd;C) and in C (Rd;C) . . . . . . . . . . . . . . . . . . . 286
b
10.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 290
11 Elliptic Equations in Rd with Homogeneous Dirichlet Boundary Condi-
+
tions 291
11.1 Solutions in H¨older Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 292
11.1.1 Further regularity results . . . . . . . . . . . . . . . . . . . . . . . . 294
11.2 Solutions in Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . 296
11.2.1 Further regularity results . . . . . . . . . . . . . . . . . . . . . . . . 304
11.3 Solutions in L∞(Rd;C) and in C (Rd;C) . . . . . . . . . . . . . . . . . . . 309
+ b +
11.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 314
12 Elliptic Equations in Rd with General Boundary Conditions 315
+
12.1 The Cα-Theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 316
12.1.1 Further regularity . . . . . . . . . . . . . . . . . . . . . . . . . . . . 319
12.2 Elliptic Equations in Lp(Rd;C) . . . . . . . . . . . . . . . . . . . . . . . . 321
+
12.2.1 Further regularity results . . . . . . . . . . . . . . . . . . . . . . . . 332
12.3 Solutions in L∞(Rd;C) and in C (Rd;C) . . . . . . . . . . . . . . . . . . . 334
+ b +
12.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 338
13 Elliptic Equations on Smooth Domains Ω 339
13.1 Elliptic Equations in Cα(Ω;C) . . . . . . . . . . . . . . . . . . . . . . . . . 339
13.1.1 Further regularity results . . . . . . . . . . . . . . . . . . . . . . . . 348
13.2 Elliptic Equations in Lp(Ω;C) . . . . . . . . . . . . . . . . . . . . . . . . . 351
13.2.1 Further regularity results . . . . . . . . . . . . . . . . . . . . . . . . 360
13.3 Solutions in L∞(Ω;C), in C(Ω;C) and in C (Ω;C) . . . . . . . . . . . . . . 364
b
13.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 368
14 Elliptic Operators and Analytic Semigroups 369
14.1 The Semigroup in C (Rd;C) . . . . . . . . . . . . . . . . . . . . . . . . . . 369
b
14.2 The Semigroups in C (Rd;C) . . . . . . . . . . . . . . . . . . . . . . . . . 374
b +
14.2.1 Proof of Theorems 7.4.1 and 7.4.3 . . . . . . . . . . . . . . . . . . . 384
14.3 The Semigroups in C (Ω;C) . . . . . . . . . . . . . . . . . . . . . . . . . . 389
b
14.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 395