(Advances in design and control) michael c delfour, jean paul zolésio shapes and geometries metrics, analysis, differential calculus, and optimization, second edition (advances in design

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Advances in Design and Control
SIAM’s Advances in Design and Control series consists of texts and monographs dealing with all areas of
design and control and their applications. Topics of interest include shape optimization, multidisciplinary
design, trajectory optimization, feedback, and optimal control. The series focuses on the mathematical and
computational aspects of engineering design and control that are usable in a wide variety of scientific and
engineering disciplines.

Editor-in-Chief
Ralph C. Smith, North Carolina State University

Editorial Board
Athanasios C. Antoulas, Rice University
Siva Banda, Air Force Research Laboratory
Belinda A. Batten, Oregon State University
John Betts, The Boeing Company (retired)
Stephen L. Campbell, North Carolina State University
Michel C. Delfour, University of Montreal
Max D. Gunzburger, Florida State University
J. William Helton, University of California, San Diego
Arthur J. Krener, University of California, Davis
Kirsten Morris, University of Waterloo
Richard Murray, California Institute of Technology
Ekkehard Sachs, University of Trier

Series Volumes

Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Metrics, Analysis, Differential Calculus, and

Optimization, Second Edition
Hovakimyan, Naira and Cao, Chengyu, L1 Adaptive Control Theory: Guaranteed Robustness with Fast Adaptation
Speyer, Jason L. and Jacobson, David H., Primer on Optimal Control Theory
Betts, John T., Practical Methods for Optimal Control and Estimation Using Nonlinear Programming, Second

Edition
Shima, Tal and Rasmussen, Steven, eds., UAV Cooperative Decision and Control: Challenges and Practical

Approaches
Speyer, Jason L. and Chung, Walter H., Stochastic Processes, Estimation, and Control
Krstic, Miroslav and Smyshlyaev, Andrey, Boundary Control of PDEs: A Course on Backstepping Designs
Ito, Kazufumi and Kunisch, Karl, Lagrange Multiplier Approach to Variational Problems and Applications
Xue, Dingyü, Chen, YangQuan, and Atherton, Derek P., Linear Feedback Control: Analysis and Design

with MATLAB
Hanson, Floyd B., Applied Stochastic Processes and Control for Jump-Diffusions: Modeling, Analysis, and

Computation
Michiels, Wim and Niculescu, Silviu-Iulian, Stability and Stabilization of Time-Delay Systems: An Eigenvalue-

Based Approach
¸ Adaptive Control Tutorial
Ioannou, Petros and Fidan, Barıs,
Bhaya, Amit and Kaszkurewicz, Eugenius, Control Perspectives on Numerical Algorithms and Matrix Problems
Robinett III, Rush D., Wilson, David G., Eisler, G. Richard, and Hurtado, John E., Applied Dynamic Programming

for Optimization of Dynamical Systems
Huang, J., Nonlinear Output Regulation: Theory and Applications

Haslinger, J. and Mäkinen, R. A. E., Introduction to Shape Optimization: Theory, Approximation, and Computation
Antoulas, Athanasios C., Approximation of Large-Scale Dynamical Systems
Gunzburger, Max D., Perspectives in Flow Control and Optimization
Delfour, M. C. and Zolésio, J.-P., Shapes and Geometries: Analysis, Differential Calculus, and Optimization
Betts, John T., Practical Methods for Optimal Control Using Nonlinear Programming
El Ghaoui, Laurent and Niculescu, Silviu-Iulian, eds., Advances in Linear Matrix Inequality Methods in Control
Helton, J. William and James, Matthew R., Extending H1 Control to Nonlinear Systems: Control of Nonlinear

Systems to Achieve Performance Objectives


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Shapes and
Geometries

Metrics, Analysis, Differential
Calculus, and Optimization
Second Edition

M. C. Delfour
Université de Montréal
Montréal, Québec
Canada

J.-P. Zolésio

National Center for Scientific Research (CNRS) and
National Institute for Research in Computer Science and Control (INRIA)
Sophia Antipolis

France

Society for Industrial and Applied Mathematics
Philadelphia


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Copyright © 2011 by the Society for Industrial and Applied Mathematics
10 9 8 7 6 5 4 3 2 1
All rights reserved. Printed in the United States of America. No part of this book may be
reproduced, stored, or transmitted in any manner without the written permission of the
publisher. For information, write to the Society for Industrial and Applied Mathematics,
3600 Market Street, 6th Floor, Philadelphia, PA 19104-2688 USA.
Trademarked names may be used in this book without the inclusion of a trademark
symbol. These names are used in an editorial context only; no infringement of trademark
is intended.
The research of the first author was supported by the Canada Council, which initiated
the work presented in this book through a Killam Fellowship; the National Sciences and
Engineering Research Council of Canada; and the FQRNT program of the Ministère de
l’Éducation du Québec.
Library of Congress Cataloging-in-Publication Data
Delfour, Michel C., 1943Shapes and geometries : metrics, analysis, differential calculus, and optimization / M. C.
Delfour, J.-P. Zolésio. -- 2nd ed.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-898719-36-9 (hardcover : alk. paper)
1. Shape theory (Topology) I. Zolésio, J.-P. II. Title.
QA612.7.D45 2011
514’.24--dc22

2010028846

is a registered trademark.


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j

This book is dedicated to
Alice, Jeanne, Jean, and Roger


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Contents
List of Figures
Preface
1
Objectives and Scope of the Book .
2
Overview of the Second Edition . .
3
Intended Audience . . . . . . . . .
4
Acknowledgments . . . . . . . . . .

xvii

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xix
. xix
. xx
. xxii
. xxiii

1 Introduction: Examples, Background, and Perspectives
1
Orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Geometry as a Variable . . . . . . . . . . . . . . . . . . . . .
1.2
Outline of the Introductory Chapter . . . . . . . . . . . . . .
2
A Simple One-Dimensional Example . . . . . . . . . . . . . . . . . .
3
Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Optimal Triangular Meshing . . . . . . . . . . . . . . . . . . . . . . .
6
Modeling Free Boundary Problems . . . . . . . . . . . . . . . . . . .
6.1
Free Interface between Two Materials . . . . . . . . . . . . .
6.2
Minimal Surfaces . . . . . . . . . . . . . . . . . . . . . . . . .
7

Design of a Thermal Diffuser . . . . . . . . . . . . . . . . . . . . . .
7.1
Description of the Physical Problem . . . . . . . . . . . . . .
7.2
Statement of the Problem . . . . . . . . . . . . . . . . . . . .
7.3
Reformulation of the Problem . . . . . . . . . . . . . . . . . .
7.4
Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . .
7.5
Design Problem . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Design of a Thermal Radiator . . . . . . . . . . . . . . . . . . . . . .
8.1
Statement of the Problem . . . . . . . . . . . . . . . . . . . .
8.2
Scaling of the Problem . . . . . . . . . . . . . . . . . . . . . .
9
A Glimpse into Segmentation of Images . . . . . . . . . . . . . . . .
9.1
Automatic Image Processing . . . . . . . . . . . . . . . . . .
9.2
Image Smoothing/Filtering by Convolution and Edge Detectors
9.2.1
Construction of the Convolution of I . . . . . . . .
9.2.2
Space-Frequency Uncertainty Relationship . . . . .
9.2.3
Laplacian Detector . . . . . . . . . . . . . . . . . . .
vii

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Contents
9.3

10

11

Objective Functions Defined on the Whole Edge . . . . . . .
9.3.1
Eulerian Shape Semiderivative . . . . . . . . . . . .
9.3.2
From Local to Global Conditions on the Edge . . .
9.4
Snakes, Geodesic Active Contours, and Level Sets . . . . . .
9.4.1
Objective Functions Defined on the Contours . . . .
9.4.2
Snakes and Geodesic Active Contours . . . . . . . .
9.4.3
Level Set Method . . . . . . . . . . . . . . . . . . .
9.4.4
Velocity Carried by the Normal . . . . . . . . . . .
9.4.5
Extension of the Level Set Equations . . . . . . . .
9.5
Objective Function Defined on the Whole Image . . . . . . .
9.5.1

Tikhonov Regularization/Smoothing . . . . . . . . .
9.5.2
Objective Function of Mumford and Shah . . . . . .
9.5.3
Relaxation of the (N − 1)-Hausdorff Measure . . . .
9.5.4
Relaxation to BV-, H s -, and SBV-Functions . . . .
9.5.5
Cracked Sets and Density Perimeter . . . . . . . . .
Shapes and Geometries: Background and Perspectives . . . . . . . .
10.1 Parametrize Geometries by Functions or Functions by
Geometries? . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.2 Shape Analysis in Mechanics and Mathematics . . . . . . . .
10.3 Characteristic Functions: Surface Measure and Geometric
Measure Theory . . . . . . . . . . . . . . . . . . . . . . . . .
10.4 Distance Functions: Smoothness, Normal, and Curvatures . .
10.5 Shape Optimization: Compliance Analysis and Sensitivity
Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.6 Shape Derivatives . . . . . . . . . . . . . . . . . . . . . . . .
10.7 Shape Calculus and Tangential Differential Calculus . . . . .
10.8 Shape Analysis in This Book . . . . . . . . . . . . . . . . . .
Shapes and Geometries: Second Edition . . . . . . . . . . . . . . . .
11.1 Geometries Parametrized by Functions . . . . . . . . . . . . .
11.2 Functions Parametrized by Geometries . . . . . . . . . . . . .
11.3 Shape Continuity and Optimization . . . . . . . . . . . . . .
11.4 Derivatives, Shape and Tangential Differential Calculuses, and
Derivatives under State Constraints . . . . . . . . . . . . . .

2 Classical Descriptions of Geometries and Their Properties
1

Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Notation and Definitions . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
Basic Notation . . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Abelian Group Structures on Subsets of a Fixed Holdall D .
2.2.1
First Abelian Group Structure on (P(D), ) . . . .
2.2.2
Second Abelian Group Structure on (P(D), ) . . .
2.3
Connected Space, Path-Connected Space, and Geodesic
Distance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Bouligand’s Contingent Cone, Dual Cone, and Normal Cone
2.5
Sobolev Spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1
Definitions . . . . . . . . . . . . . . . . . . . . . . .

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Contents

ix

2.6

3

4
5

Sets
3.1
3.2

3.3
Sets
Sets
5.1
5.2
5.3
5.4

6

Sets
6.1
6.2
6.3
6.4

6.5

2.5.2
The Space W0m,p (Ω) . . . . . . . . . . . . . . . . . . 61
2.5.3
Embedding of H01 (Ω) into H01 (D) . . . . . . . . . . 62
2.5.4
Projection Operator . . . . . . . . . . . . . . . . . . 63
Spaces of Continuous and Differentiable Functions . . . . . . 63
2.6.1
Continuous and C k Functions . . . . . . . . . . . . 63
2.6.2
Hă
older (C 0, ) and Lipschitz (C 0,1 ) Continuous
Functions . . . . . . . . . . . . . . . . . . . . . . . . 65
2.6.3
Embedding Theorem . . . . . . . . . . . . . . . . . 65
2.6.4
Identity C k,1 (Ω) = W k+1,∞ (Ω): From Convex to
Path-Connected Domains via the Geodesic Distance 66
Locally Described by an Homeomorphism or a Diffeomorphism
67
Sets of Classes C k and C k, . . . . . . . . . . . . . . . . . . . 67
Boundary Integral, Canonical Density, and Hausdorff Measures 70

3.2.1
Boundary Integral for Sets of Class C 1 . . . . . . . 70
3.2.2
Integral on Submanifolds . . . . . . . . . . . . . . . 71
3.2.3
Hausdorff Measures . . . . . . . . . . . . . . . . . . 72
Fundamental Forms and Principal Curvatures . . . . . . . . . 73
Globally Described by the Level Sets of a Function . . . . . . . 75
Locally Described by the Epigraph of a Function . . . . . . . . 78
Local C 0 Epigraphs, C 0 Epigraphs, and Equi-C 0 Epigraphs
and the Space H of Dominating Functions . . . . . . . . . . . 79
olderian/Lipschitzian Sets . . . . 87
Local C k, -Epigraphs and Hă
Local C k, -Epigraphs and Sets of Class C k, . . . . . . . . . . 89
Locally Lipschitzian Sets: Some Examples and Properties . . 92
5.4.1
Examples and Continuous Linear Extensions . . . . 92
5.4.2
Convex Sets . . . . . . . . . . . . . . . . . . . . . . 93
5.4.3
Boundary Measure and Integral for Lipschitzian Sets 94
5.4.4
Geodesic Distance in a Domain and in Its Boundary 97
5.4.5
Nonhomogeneous Neumann and Dirichlet Problems 100
Locally Described by a Geometric Property . . . . . . . . . . . 101
Definitions and Main Results . . . . . . . . . . . . . . . . . . 102
Equivalence of Geometric Segment and C 0 Epigraph
Properties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Equivalence of the Uniform Fat Segment and the Equi-C 0

Epigraph Properties . . . . . . . . . . . . . . . . . . . . . . . 109
Uniform Cone/Cusp Properties and Hă
olderian/Lipschitzian
Sets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.4.1
Uniform Cone Property and Lipschitzian Sets . . . 114
6.4.2
Uniform Cusp Property and Hă
olderian Sets . . . . . 115
Hausdorff Measure and Dimension of the Boundary . . . . . 116

3 Courant Metrics on Images of a Set
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Generic Constructions of Micheletti . . . . . . . . . . . . .
2.1
Space F(Θ) of Transformations of RN . . . . . . .
2.2
Diffeomorphisms for B(RN , RN ) and C0∞ (RN , RN )

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x

Contents
Closed Subgroups G . . . . . . . . . . . . . . . . . . . . . . .
Courant Metric on the Quotient Group F(Θ)/G . . . . . . .
Assumptions for B k (RN , RN ), C k (RN , RN ), and C0k (RN , RN )
2.5.1
Checking the Assumptions . . . . . . . . . . . . . .
2.5.2
Perturbations of the Identity and Tangent Space . .
2.6
Assumptions for C k,1 (RN , RN ) and C0k,1 (RN , RN ) . . . . . .
2.6.1
Checking the Assumptions . . . . . . . . . . . . . .
2.6.2
Perturbations of the Identity and Tangent Space . .
Generalization to All Homeomorphisms and C k -Diffeomorphisms . .

2.3
2.4
2.5

3

4 Transformations Generated by Velocities
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Metrics on Transformations Generated by Velocities . . . . . . .
2.1

Subgroup GΘ of Transformations Generated by Velocities
2.2
Complete Metrics on GΘ and Geodesics . . . . . . . . . .
2.3
Constructions of Azencott and Trouv´e . . . . . . . . . . .
3
Semiderivatives via Transformations Generated by Velocities . .
3.1
Shape Function . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Gateaux and Hadamard Semiderivatives . . . . . . . . . .
3.3
Examples of Families of Transformations of Domains . . .
3.3.1
C ∞ -Domains . . . . . . . . . . . . . . . . . . . .
3.3.2
C k -Domains . . . . . . . . . . . . . . . . . . . .
3.3.3
Cartesian Graphs . . . . . . . . . . . . . . . . .
3.3.4
Polar Coordinates and Star-Shaped Domains . .
3.3.5
Level Sets . . . . . . . . . . . . . . . . . . . . . .
4
Unconstrained Families of Domains . . . . . . . . . . . . . . . . .
4.1
Equivalence between Velocities and Transformations . . .
4.2
Perturbations of the Identity . . . . . . . . . . . . . . . .
4.3

Equivalence for Special Families of Velocities . . . . . . .
5
Constrained Families of Domains . . . . . . . . . . . . . . . . . .
5.1
Equivalence between Velocities and Transformations . . .
5.2
Transformation of Condition (V2D ) into a Linear
Constraint . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Continuity of Shape Functions along Velocity Flows . . . . . . .
5 Metrics via Characteristic Functions
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Abelian Group Structure on Measurable Characteristic Functions
2.1
Group Structure on Xµ (RN ) . . . . . . . . . . . . . . . .
2.2
Measure Spaces . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Complete Metric for Characteristic Functions in
Lp -Topologies . . . . . . . . . . . . . . . . . . . . . . . . .
3
Lebesgue Measurable Characteristic Functions . . . . . . . . . .
3.1
Strong Topologies and C ∞ -Approximations . . . . . . . .
3.2
Weak Topologies and Microstructures . . . . . . . . . . .
3.3
Nice or Measure Theoretic Representative . . . . . . . . .

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5
6

7

xi
3.4
The Family of Convex Sets . . . . . . . . . . . . . . . . . . .
3.5
Sobolev Spaces for Measurable Domains . . . . . . . . . . . .
Some Compliance Problems with Two Materials . . . . . . . . . . .
4.1
Transmission Problem and Compliance . . . . . . . . . . . .
4.2
The Original Problem of C´ea and Malanowski . . . . . . . . .
4.3
Relaxation and Homogenization . . . . . . . . . . . . . . . .
Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . . .
Caccioppoli or Finite Perimeter Sets . . . . . . . . . . . . . . . . . .
6.1
Finite Perimeter Sets . . . . . . . . . . . . . . . . . . . . . . .
6.2
Decomposition of the Integral along Level Sets . . . . . . . .
6.3
Domains of Class W ε,p (D), 0 ≤ ε < 1/p, p ≥ 1, and a Cascade
of Complete Metric Spaces . . . . . . . . . . . . . . . . . . .
6.4
Compactness and Uniform Cone Property . . . . . . . . . . .
Existence for the Bernoulli Free Boundary Problem . . . . . . . . . .
7.1
An Example: Elementary Modeling of the Water Wave . . .
7.2

Existence for a Class of Free Boundary Problems . . . . . . .
7.3
Weak Solutions of Some Generic Free Boundary Problems . .
7.3.1
Problem without Constraint . . . . . . . . . . . . .
7.3.2
Constraint on the Measure of the Domain Ω . . . .
7.4
Weak Existence with Surface Tension . . . . . . . . . . . . .

6 Metrics via Distance Functions
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Uniform Metric Topologies . . . . . . . . . . . . . . . . . . . . . . .
2.1
Family of Distance Functions Cd (D) . . . . . . . . . . . . . .
2.2
Pomp´eiu–Hausdorff Metric on Cd (D) . . . . . . . . . . . . . .
2.3
Uniform Complementary Metric Topology and Cdc (D) . . . .
c
2.4
Families Cdc (E; D) and Cd,loc
(E; D) . . . . . . . . . . . . . . .
3
Projection, Skeleton, Crack, and Differentiability . . . . . . . . . . .
4
W 1,p -Metric Topology and Characteristic Functions . . . . . . . . .
4.1

Motivations and Main Properties . . . . . . . . . . . . . . . .
4.2
Weak W 1,p -Topology . . . . . . . . . . . . . . . . . . . . . . .
5
Sets of Bounded and Locally Bounded Curvature . . . . . . . . . . .
5.1
Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6
Reach and Federer’s Sets of Positive Reach . . . . . . . . . . . . . .
6.1
Definitions and Main Properties . . . . . . . . . . . . . . . .
6.2
C k -Submanifolds . . . . . . . . . . . . . . . . . . . . . . . . .
6.3
A Compact Family of Sets with Uniform Positive Reach . . .
7
Approximation by Dilated Sets/Tubular Neighborhoods and Critical
Points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8
Characterization of Convex Sets . . . . . . . . . . . . . . . . . . . .
8.1
Convex Sets and Properties of dA . . . . . . . . . . . . . . . .
8.2
Semiconvexity and BV Character of dA . . . . . . . . . . . .
8.3
Closed Convex Hull of A and Fenchel Transform of dA . . . .
8.4
Families of Convex Sets Cd (D), Cdc (D), Cdc (E; D), and
c
(E; D) . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Cd,loc

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9

Compactness Theorems for Sets of Bounded Curvature . . . . . . . . 324
9.1
Global Conditions in D . . . . . . . . . . . . . . . . . . . . . 325
9.2
Local Conditions in Tubular Neighborhoods . . . . . . . . . . 327

7 Metrics via Oriented Distance Functions

1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Uniform Metric Topology . . . . . . . . . . . . . . . . . . . . . . . .
2.1
The Family of Oriented Distance Functions Cb (D) . . . . . .
2.2
Uniform Metric Topology . . . . . . . . . . . . . . . . . . . .
3
Projection, Skeleton, Crack, and Differentiability . . . . . . . . . . .
4
W 1,p (D)-Metric Topology and the Family Cb0 (D) . . . . . . . . . . .
4.1
Motivations and Main Properties . . . . . . . . . . . . . . . .
4.2
Weak W 1,p -Topology . . . . . . . . . . . . . . . . . . . . . . .
5
Boundary of Bounded and Locally Bounded Curvature . . . . . . . .
5.1
Examples and Limit of Tubular Norms as h Goes to Zero . .
6
Approximation by Dilated Sets/Tubular Neighborhoods . . . . . . .
7
Federer’s Sets of Positive Reach . . . . . . . . . . . . . . . . . . . . .
7.1
Approximation by Dilated Sets/Tubular Neighborhoods . . .
7.2
Boundaries with Positive Reach . . . . . . . . . . . . . . . . .
8
Boundary Smoothness and Smoothness of bA . . . . . . . . . . . . .

9
Sobolev or W m,p Domains . . . . . . . . . . . . . . . . . . . . . . . .
10 Characterization of Convex and Semiconvex Sets . . . . . . . . . . .
10.1 Convex Sets and Convexity of bA . . . . . . . . . . . . . . . .
10.2 Families of Convex Sets Cb (D), Cb (E; D), and
Cb,loc (E; D) . . . . . . . . . . . . . . . . . . . . . . . . . . . .
10.3 BV Character of bA and Semiconvex Sets . . . . . . . . . . .
11 Compactness and Sets of Bounded Curvature . . . . . . . . . . . . .
11.1 Global Conditions on D . . . . . . . . . . . . . . . . . . . . .
11.2 Local Conditions in Tubular Neighborhoods . . . . . . . . . .
12 Finite Density Perimeter and Compactness . . . . . . . . . . . . . .
13 Compactness and Uniform Fat Segment Property . . . . . . . . . . .
13.1 Main Theorem . . . . . . . . . . . . . . . . . . . . . . . . . .
13.2 Equivalent Conditions on the Local Graph Functions . . . . .
14 Compactness under the Uniform Fat Segment Property and a Bound
on a Perimeter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
14.1 De Giorgi Perimeter of Caccioppoli Sets . . . . . . . . . . . .
14.2 Finite Density Perimeter . . . . . . . . . . . . . . . . . . . . .
15 The Families of Cracked Sets . . . . . . . . . . . . . . . . . . . . . .
16 A Variation of the Image Segmentation Problem of Mumford
and Shah . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
16.1 Problem Formulation . . . . . . . . . . . . . . . . . . . . . .
16.2 Cracked Sets without the Perimeter . . . . . . . . . . . . . .
16.2.1
Technical Lemmas . . . . . . . . . . . . . . . . . . .
16.2.2
Another Compactness Theorem . . . . . . . . . . .
16.2.3
Proof of Theorem 16.1 . . . . . . . . . . . . . . . . .
16.3 Existence of a Cracked Set with Minimum Density Perimeter

335
335
337
337
339
344
349
349
352
354
355
358
361
361
363
365
373
375
375
379
380
381
382
382
385
387
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391
393

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16.4

Uniform Bound or Penalization Term in the Objective
Function on the Density Perimeter . . . . . . . . . . . . . . . 407

8 Shape Continuity and Optimization
1
Introduction and Generic Examples . . . . . . . . . . . . . . . . . . .
1.1
First Generic Example . . . . . . . . . . . . . . . . . . . . . .
1.2
Second Generic Example . . . . . . . . . . . . . . . . . . . . .
1.3

Third Generic Example . . . . . . . . . . . . . . . . . . . . .
1.4
Fourth Generic Example . . . . . . . . . . . . . . . . . . . . .
2
Upper Semicontinuity and Maximization of the First Eigenvalue . .
3
Continuity of the Transmission Problem . . . . . . . . . . . . . . . .
4
Continuity of the Homogeneous Dirichlet Boundary Value Problem .
4.1
Classical, Relaxed, and Overrelaxed Problems . . . . . . . . .
4.2
Classical Dirichlet Boundary Value Problem . . . . . . . . . .
4.3
Overrelaxed Dirichlet Boundary Value Problem . . . . . . . .
4.3.1
Approximation by Transmission Problems . . . . . .
4.3.2
Continuity with Respect to X(D) in the
Lp (D)-Topology . . . . . . . . . . . . . . . . . . . .
4.4
Relaxed Dirichlet Boundary Value Problem . . . . . . . . . .
5
Continuity of the Homogeneous Neumann Boundary Value Problem
6
Elements of Capacity Theory . . . . . . . . . . . . . . . . . . . . . .
6.1
Definition and Basic Properties . . . . . . . . . . . . . . . . .
6.2
Quasi-continuous Representative and H 1 -Functions . . . . . .

6.3
Transport of Sets of Zero Capacity . . . . . . . . . . . . . . .
7
Crack-Free Sets and Some Applications . . . . . . . . . . . . . . . .
7.1
Definitions and Properties . . . . . . . . . . . . . . . . . . . .
7.2
Continuity and Optimization over L(D, r, O, λ) . . . . . . . .
7.2.1
Continuity of the Classical Homogeneous Dirichlet
Boundary Condition . . . . . . . . . . . . . . . . . .
7.2.2
Minimization/Maximization of the First
Eigenvalue . . . . . . . . . . . . . . . . . . . . . . .
8
Continuity under Capacity Constraints . . . . . . . . . . . . . . . . .
9
Compact Families Oc,r (D) and Lc,r (O, D) . . . . . . . . . . . . . . .
9.1
Compact Family Oc,r (D) . . . . . . . . . . . . . . . . . . . .
9.2
Compact Family Lc,r (O, D) and Thick Set Property . . . . .
9.3
Maximizing the Eigenvalue λA (Ω) . . . . . . . . . . . . . . .
9.4
State Constrained Minimization Problems . . . . . . . . . . .
9.5
Examples with a Constraint on the Gradient . . . . . . . . .

409

409
411
411
411
412
412
417
418
418
421
423
423

9 Shape and Tangential Differential Calculuses
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . .
2
Review of Differentiation in Topological Vector Spaces
2.1
Definitions of Semiderivatives and Derivatives .
2.2
Derivatives in Normed Vector Spaces . . . . . .
2.3
Locally Lipschitz Functions . . . . . . . . . . .
2.4
Chain Rule for Semiderivatives . . . . . . . . .

457
457
458

458
461
465
465

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xiv

Contents

3

4

5

6

2.5
Semiderivatives of Convex Functions . . . . . . . . . . .
2.6
Hadamard Semiderivative and Velocity Method . . . . .
First-Order Shape Semiderivatives and Derivatives . . . . . . .
3.1
Eulerian and Hadamard Semiderivatives . . . . . . . . .
3.2
Hadamard Semidifferentiability and Courant Metric
Continuity . . . . . . . . . . . . . . . . . . . . . . . . .
3.3
Perturbations of the Identity and Gateaux and Fr´echet
Derivatives . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Shape Gradient and Structure Theorem . . . . . . . . .
Elements of Shape Calculus . . . . . . . . . . . . . . . . . . . .
4.1
Basic Formula for Domain Integrals . . . . . . . . . . .
4.2
Basic Formula for Boundary Integrals . . . . . . . . . .
4.3
Examples of Shape Derivatives . . . . . . . . . . . . . .
4.3.1
Volume of Ω and Surface Area of Γ . . . . . .

4.3.2
H 1 (Ω)-Norm . . . . . . . . . . . . . . . . . . .
4.3.3
Normal Derivative . . . . . . . . . . . . . . . .
Elements of Tangential Calculus . . . . . . . . . . . . . . . . .
5.1
Intrinsic Definition of the Tangential Gradient . . . . .
5.2
First-Order Derivatives . . . . . . . . . . . . . . . . . .
5.3
Second-Order Derivatives . . . . . . . . . . . . . . . . .
5.4
A Few Useful Formulae and the Chain Rule . . . . . . .
5.5
The Stokes and Green Formulae . . . . . . . . . . . . .
5.6
Relation between Tangential and Covariant Derivatives
5.7
Back to the Example of Section 4.3.3 . . . . . . . . . . .
Second-Order Semiderivative and Shape Hessian . . . . . . . .
6.1
Second-Order Derivative of the Domain Integral . . . .
6.2
Basic Formula for Domain Integrals . . . . . . . . . . .
6.3
Nonautonomous Case . . . . . . . . . . . . . . . . . . .
6.4
Autonomous Case . . . . . . . . . . . . . . . . . . . . .
6.5
Decomposition of d2 J(Ω; V (0), W (0)) . . . . . . . . . .

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469
471
471

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10 Shape Gradients under a State Equation Constraint
1
Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
Min Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1
An Illustrative Example and a Shape Variational Principle
2.2
Function Space Parametrization . . . . . . . . . . . . . . .
2.3
Differentiability of a Minimum with Respect to a
Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
Application of the Theorem . . . . . . . . . . . . . . . . . .
2.5
Domain and Boundary Integral Expressions of the Shape
Gradient . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3
Buckling of Columns . . . . . . . . . . . . . . . . . . . . . . . . . .
4

Eigenvalue Problems . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1
Transport of H0k (Ω) by W k,∞ -Transformations of RN . . .
4.2
Laplacian and Bi-Laplacian . . . . . . . . . . . . . . . . . .
4.3
Linear Elasticity . . . . . . . . . . . . . . . . . . . . . . . .

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476
479
482
482
484
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486
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488
491
492
495
496
497
498
498
501
501
502
504
505
510
515

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519
521
521
522

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6

xv
Saddle
5.1
5.2
5.3
5.4

Point Formulation and Function Space Parametrization . .
An Illustrative Example . . . . . . . . . . . . . . . . . . . .
Saddle Point Formulation . . . . . . . . . . . . . . . . . . .
Function Space Parametrization . . . . . . . . . . . . . . .
Differentiability of a Saddle Point with Respect to a
Parameter . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.5
Application of the Theorem . . . . . . . . . . . . . . . . . .
5.6
Domain and Boundary Expressions for the Shape Gradient
Multipliers and Function Space Embedding . . . . . . . . . . . . .
6.1
The Nonhomogeneous Dirichlet Problem . . . . . . . . . . .
6.2
A Saddle Point Formulation of the State Equation . . . . .
6.3
Saddle Point Expression of the Objective Function . . . . .
6.4
Verification of the Assumptions of Theorem 5.1 . . . . . . .

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551
552
553

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559
561
562
562
563
564
566

Elements of Bibliography

571

Index of Notation

615

Index

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List of Figures
1.1
1.2
1.3
1.4
1.5
1.6

Graph of J(a). . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Column of height one and cross section area A under the load .
Triangulation and basis function associated with node Mi . . . . .
Fixed domain D and its partition into Ω1 and Ω2 . . . . . . . . .
Heat spreading scheme for high-power solid-state devices. . . . .
(A) Volume Ω and its boundary Σ; (B) Surface A generating Ω;
(C) Surface D generating Ω. . . . . . . . . . . . . . . . . . . . . .
1.7 Volume Ω and its cross section. . . . . . . . . . . . . . . . . . . .

1.8 Volume Ω and its generating surface A. . . . . . . . . . . . . . .
1.9 Image I of objects and their segmentation in the frame D. . . . .
1.10 Image I containing black curves or cracks in the frame D. . . . .
1.11 Example of a two-dimensional strongly cracked set. . . . . . . . .
1.12 Example of a surface with facets associated with a ball. . . . . .

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5
8
11
14

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19
20
22
22
35
37
68
79
91
92

2.8

Diffeomorphism gx from U (x) to B. . . . . . . . . . . . . . . . . . .
Local epigraph representation (N = 2). . . . . . . . . . . . . . . . . .
Domain Ω0 and its image T (Ω0 ) spiraling around the origin. . . . . .
Domain Ω0 and its image T (Ω0 ) zigzagging towards the origin. . . .
Examples of arbitrary and axially symmetrical O around the
direction d = Ax (0, eN ). . . . . . . . . . . . . . . . . . . . . . . . . .

The cone x + Ax C(λ, ω) in the direction Ax eN . . . . . . . . . . . . .
Domain Ω for N = 2, 0 < α < 1, e2 = (0, 1), ρ = 1/6, λ = (1/6)α ,
h(θ) = θ α . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
f (x) = dC (x)1/2 constructed on the Cantor set C for 2k + 1 = 3. . .

4.1

Transport of Ω by the velocity field V . . . . . . . . . . . . . . . . . . 171

5.1
5.2
5.3
5.4
5.5

Smiling sun Ω and expressionless sun Ω. . . . . . . . . . . . . .
Disconnected domain Ω = Ω0 ∪ Ω1 ∪ Ω2 . . . . . . . . . . . . . .
Fixed domain D and its partition into Ω1 and Ω2 . . . . . . . .
The function f (x, y) = 56 (1 − |x| − |y|)6 . . . . . . . . . . . . .
Optimal distribution and isotherms with k1 = 2 (black) and k2
(white) for the problem of section 4.1. . . . . . . . . . . . . . .
Optimal distribution and isotherms with k1 = 2 (black) and k2
(white) for the problem of C´ea and Malanowski. . . . . . . . .

2.1
2.2
2.3
2.4
2.5
2.6

2.7

5.6

xvii

. . .
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=1
. . .
=1
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109
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118
119

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xviii

List of Figures

5.7

The staircase. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 248

6.1
6.2
6.3
6.4
6.5

Skeletons Sk (A), Sk ( A), and Sk (∂A) = Sk (A) ∪ Sk (
Nonuniqueness of the exterior normal. . . . . . . . . .
Vertical stripes of Example 4.1. . . . . . . . . . . . . .
∇dA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . .
Set of critical points of A. . . . . . . . . . . . . . . . .

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280
286
293
301

318

7.1
7.2

∇bA for Examples 5.1, 5.2, and 5.3. . . . . . . . . . . . . . . . . . . .
W 1,p -convergence of a sequence of open subsets {An : n ≥ 1} of R2
with uniformly bounded density perimeter to a set with empty
interior. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Example of a two-dimensional strongly cracked set. . . . . . . . . . .
The two-dimensional strongly cracked set of Figure 7.3 in an open
frame D. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
The two open components Ω1 and Ω2 of the open domain Ω for
N = 2. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

356

7.3
7.4
7.5

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Preface

1

Objectives and Scope of the Book

The objective of this book is to give a comprehensive presentation of mathematical
constructions and tools that can be used to study problems where the modeling,
optimization, or control variable is no longer a set of parameters or functions but
the shape or the structure of a geometric object. In that context, a good analytical
framework and good modeling techniques must be able to handle the occurrence of
singular behaviors whenever they are compatible with the mechanics or the physics
of the problems at hand. In some optimization problems, the natural intuitive
notion of a geometric domain undergoes mutations into relaxed entities such as
microstructures. So the objects under consideration need not be smooth open domains, or even sets, as long as they still makes sense mathematically.
This book covers the basic mathematical ideas, constructions, and methods
that come from different fields of mathematical activities and areas of applications
that have often evolved in parallel directions. The scope of research is frighteningly
broad because it touches on areas that include classical geometry, modern partial differential equations, geometric measure theory, topological groups, and constrained
optimization, with applications to classical mechanics of continuous media such as
fluid mechanics, elasticity theory, fracture theory, modern theories of optimal design, optimal location and shape of geometric objects, free and moving boundary
problems, and image processing. Innovative modeling or new issues raised in some
applications force a new look at the fundamentals of well-established mathematical
areas such as geometry, to relax basic notions of volume, perimeter, and curvature
or boundary value problems, and to find suitable relaxations of solutions. In that
spirit, Henri Lebesgue was probably a pioneer when he relaxed the intuitive notion
of volume to the one of measure on an equivalence class of measurable sets in 1907.
He was followed in that endeavor in the early 1950s by the celebrated work of E. De
Giorgi, who used the relaxed notion of perimeter defined on the class of Caccioppoli
sets to solve Plateau’s problem of minimal surfaces.
The material that is pertinent to the study of geometric objects and the entities and functions that are defined on them would necessitate an encyclopedic

investment to bring together the basic theories and their fields of applications. This
objective is obviously beyond the scope of a single book and two authors. The
xix


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xx

Preface

coverage of this book is more modest. Yet, it contains most of the important fundamentals at this stage of evolution of this expanding field.
Even if shape analysis and optimization have undergone considerable and important developments on the theoretical and numerical fronts, there are still cultural
barriers between areas of applications and between theories. The whole field is extremely active, and the best is yet to come with fundamental structures and tools
beginning to emerge. It is hoped that this book will help to build new bridges and
stimulate cross-fertilization of ideas and methods.

2

Overview of the Second Edition

The second edition is almost a new book. All chapters from the first edition have
been updated and, in most cases, considerably enriched with new material. Many
chapters or parts of chapters have been completely rewritten following the developments in the field over the past 10 years. The book went from 9 to 10 chapters
with a more elaborate sectioning of each chapter in order to produce a much more
detailed table of contents. This makes it easier to find specific material.
A series of illustrative generic examples has been added right at the beginning of the introductory Chapter 1 to motivate the reader and illustrate the basic
dilemma: parametrize geometries by functions or functions by geometries? This is
followed by the big picture: a section on background and perspectives and a more
detailed presentation of the second edition.

The former Chapter 2 has been split into Chapter 2 on the classical descriptions and properties of domains and sets and a new Chapter 3, where the important
material on Courant metrics and the generic constructions of A. M. Micheletti have
been reorganized and expanded. Basic definitions and material have been added
and regrouped at the beginning of Chapter 2: Abelian group structure on subsets
of a set, connected and path-connected spaces, function spaces, tangent and dual
cones, and geodesic distance. The coverage of domains that verify some segment
property and have a local epigraph representation has been considerably expanded,
and Lipschitzian (graph) domains are now dealt with as a special case.
The new Chapter 3 on domains and submanifolds that are the image of a
fixed set considerably expands the material of the first edition by bringing up the
general assumptions behind the generic constructions of A. M. Micheletti that lead
to the Courant metrics on the quotient space of families of transformations by
subgroups of isometries such as identities, rotations, translations, or flips. The
general results apply to a broad range of groups of transformations of the Euclidean
space and to arbitrary closed subgroups. New complete metrics on the whole spaces
of homeomorphisms and C k -diffeomorphisms are also introduced to extend classical
results for transformations of compact manifolds to general unbounded closed sets
and open sets that are crack-free. This material is central in classical mechanics
and physics and in modern applications such as imaging and detection.
The former Chapter 7 on transformations versus flows of velocities has been
moved right after the Courant metrics as Chapter 4 and considerably expanded.
It now specializes the results of Chapter 3 to spaces of transformations that are


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2. Overview of the Second Edition

xxi

generated by the flow of a velocity field over a generic time interval. One important
motivation is to introduce a notion of semiderivatives as well as a tractable criterion
for continuity with respect to Courant metrics. Another motivation for the velocity
point of view is the general framework of R. Azencott and A. Trouv´e starting in
1994 with applications in imaging. They construct complete metrics in relation
with geodesic paths in spaces of diffeomorphisms generated by a velocity field.
The former Chapter 3 on the relaxation to measurable sets and Chapters 4 and
5 on distance and oriented distance functions have become Chapters 5, 6, and 7.
Those chapters have been renamed Metrics Generated by . . . in order to emphasize
one of the main thrusts of the book: the construction of complete metrics on shapes
and geometries.1 Those chapters emphasize the function analytic description of sets
and domains: construction of metric topologies and characterization of compact
families of sets or submanifolds in the Euclidean space. In that context, we are now
dealing with equivalence classes of sets that may or may not have an invariant open
or closed representative in the class. For instance, they include Lebesgue measurable
sets and Federer’s sets of positive reach. Many of the classical properties of sets can
be recovered from the smoothness or function analytic properties of those functions.
The former Chapter 6 on optimization of shape functions has been completely
rewritten and expanded as Chapter 8 on shape continuity and optimization. With
meaningful metric topologies, we can now speak of continuity of a geometric objective functional such as the volume, the perimeter, the mean curvature, etc., compact
families of sets, and existence of optimal geometries. The chapter concentrates on
continuity issues related to shape optimization problems under state equation constraints. A special family of state constrained problems are the ones for which the
objective function is defined as an infimum over a family of functions over a fixed
domain or set such as the eigenvalue problems. We first characterize the continuity
of the transmission problem and the upper semicontinuity of the first eigenvalue of
the generalized Laplacian with respect to the domain. We then study the continuity of the solution of the homogeneous Dirichlet and Neumann boundary value
problems with respect to their underlying domain of definition since they require
different constructions and topologies that are generic of the two types of boundary
conditions even for more complex nonlinear partial differential equations. An introduction is also given to the concepts and results from capacity theory from which
very general families of sets stable with respect to boundary conditions can be constructed. Note that some material has been moved from one chapter to another.

For instance, section 7 on the continuity of the Dirichlet boundary problem in the
former Chapter 3 has been merged with the content of the former Chapter 4 in the
new Chapter 8.
The former Chapters 8 and 9 have become Chapters 9 and 10. They are
devoted to a modern version of the shape calculus, an introduction to the tangential
differential calculus, and the shape derivatives under a state equation constraint. In
Chapters 3, 5, 6, and 7, we have constructed complete metric spaces of geometries.
Those spaces are nonlinear and nonconvex. However, several of them have a group
1 This is in line with current trends in the literature such as in the work of the 2009 Abel Prize
´moli and G.
winner M. Gromov [1] and its applications in imaging by G. Sapiro [1] and F. Me
Sapiro [1] to identify objects up to an isometry.


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xxii

Preface

structure and, in some cases, it is possible to construct C 1 -paths in the group
from velocity fields. This leads to the notion of Eulerian semiderivative that is
somehow the analogue of a derivative on a smooth manifold. In fact, two types
of semiderivatives are of interest: the weaker Gateaux style semiderivative and the
stronger Hadamard style semiderivative. In the latter case, the classical chain rule
is still available even for nondifferentiable functions. In order to prepare the ground
for shape derivatives, an enriched self-contained review of the pertinent material on
semiderivatives and derivatives in topological vector spaces is provided.
The important Chapter 10 concentrates on two generic examples often encountered in shape optimization. The first one is associated with the so-called compliance
problems, where the shape functional is itself the minimum of a domain-dependent

energy functional. The special feature of such functionals is that the adjoint state
coincides with the state. This obviously leads to considerable simplifications in the
analysis. In that case, it will be shown that theorems on the differentiability of
the minimum of a functional with respect to a real parameter readily give explicit
expressions of the Eulerian semiderivative even when the minimizer is not unique.
The second one will deal with shape functionals that can be expressed as the saddle
point of some appropriate Lagrangian. As in the first example, theorems on the
differentiability of the saddle point of a functional with respect to a real parameter
readily give explicit expressions of the Eulerian semiderivative even when the solution of the saddle point equations is not unique. Avoiding the differentiation of
the state equation with respect to the domain is particularly advantageous in shape
problems.

3

Intended Audience

The targeted audience is applied mathematicians and advanced engineers and scientists, but the book is also suitable for a broader audience of mathematicians as a
relatively well-structured initiation to shape analysis and calculus techniques. Some
of the chapters are fairly self-contained and of independent interest. They can be
used as lecture notes for a mini-course. The material at the beginning of each
chapter is accessible to a broad audience, while the latter sections may sometimes
require more mathematical maturity. Thus the book can be used as a graduate text
as well as a reference book. It complements existing books that emphasize specific
mechanical or engineering applications or numerical methods. It can be considered
´sio [9], Introduction
a companion to the book of J. Sokolowski and J.-P. Zole
to Shape Optimization, published in 1992.
Earlier versions of parts of this book have been used as lecture notes in graduate courses at the Universit´e de Montr´eal in 1986–1987, 1993–1994, 1995–1996,
and 1997–1998 and at international meetings, workshops, or schools: S´eminaire de
Math´ematiques Sup´erieures on Shape Optimization and Free Boundaries (Montr´eal,

Canada, June 25 to July 13, 1990), short course on Shape Sensitivity Analysis
(K´enitra, Morocco, December 1993), course of the COMETT MATARI European
Program on Shape Optimization and Mutational Equations (Sophia-Antipolis,
France, September 27 to October 1, 1993), CRM Summer School on Boundaries,


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4. Acknowledgments

xxiii

Interfaces and Transitions (Banff, Canada, August 6–18, 1995), and CIME course
on Optimal Design (Troia, Portugal, June 1998).

4

Acknowledgments

The first author is pleased to acknowledge the support of the Canada Council, which
initiated the work presented in this book through a Killam Fellowship; the constant
support of the National Sciences and Engineering Research Council of Canada;
´
and the FQRNT program of the Minist`ere de l’Education
du Qu´ebec. Many
thanks also to Louise Letendre and Andr´e Montpetit of the Centre de Recherches
Math´ematiques, who provided their technical support, experience, and talent over
the extended period of gestation of this book.
Michel Delfour
Jean-Paul Zol´esio

August 13, 2009


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Chapter 1

Introduction: Examples,
Background, and
Perspectives
1
1.1

Orientation
Geometry as a Variable

The central object of this book1 is the geometry as a variable. As in the theory
of functions of real variables, we need a differential calculus, spaces of geometries,
evolution equations, and other familiar concepts in analysis when the variable is
no longer a scalar, a vector, or a function, but is a geometric domain. This is
motivated by many important problems in science and engineering that involve the
geometry as a modeling, design, or control variable. In general the geometric objects
we shall consider will not be parametrized or structured. Yet we are not starting
from scratch, and several building blocks are already available from many fields:
geometric measure theory, physics of continuous media, free boundary problems,
the parametrization of geometries by functions, the set derivative as the inverse of
the integral, the parametrization of functions by geometries, the Pomp´eiu–Hausdorff
metric, and so on.
As is often the case in mathematics, spaces of geometries and notions of derivatives with respect to the geometry are built from well-established elements of
functional analysis and differential calculus. There are many ways to structure

families of geometries. For instance, a domain can be made variable by considering
1 The numbering of equations, theorems, lemmas, corollaries, definitions, examples, and remarks
is by chapter. When a reference to another chapter is necessary it is always followed by the words
in Chapter and the number of the chapter. For instance, “equation (2.18) in Chapter 9.” The
text of theorems, lemmas, and corollaries is slanted; the text of definitions, examples, and remarks
is normal shape and ended by a square
. This makes it possible to aesthetically emphasize
certain words especially in definitions. The bibliography is by author in alphabetical order. For
each author or group of coauthors, there is a numbering in square brackets starting with [1]. A
´ [3] and a reference to an
reference to an item by a single author is of the form J. Dieudonne
item with several coauthors S. Agmon, A. Douglis, and L. Nirenberg [2]. Boxed formulae or
statements are used in some chapters for two distinct purposes. First, they emphasize certain
important definitions, results, or identities; second, in long proofs of some theorems, lemmas, or
corollaries, they isolate key intermediary results which will be necessary to more easily follow the
subsequent steps of the proof.

1