Mathematical Analysis: Foundations and Advanced Techniques for Functions of Several Variables

418 Pages·2012·5.11 MB·English

by  Mariano Giaquinta

Mariano Giaquinta • Giuseppe Modica
Mathematical Analysis
Foundations and Advanced Techniques
for Functions of Several Variables
Mariano Giaquinta Giuseppe Modica
Scuola Normale Superiore Dipartimento di Sistemi e Informatica
Piazza dei Cavalieri, 7 Università di Firenze
I-56100 Pisa Via S. Marta, 3
Italy I-50139 Firenze
[e-mail] Italy
[e-mail]
ISBN 978-0-8176-8309-2 e-ISBN 978-0-8176-8310-8
DOI 10.1007/978-0-8176-8310-8
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2011940806
Mathematics Subject Classification: 28-01, 35-01, 49-01, 52-01, 58A10

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Preface
Thisvolume1,thataddstothefourvolumes2 thatalreadyappeared,com-
plements the study of ideas and techniques of the differential and integral
calculus for functions of several variables with the presentation of several
specific topics of particular relevance from which the calculus of functions
of several variables has originated and in which it has its most natural
context. Some chapters have to be seen as introductory to further de-
velopments that proceed autonomously and that cannot be treated here
because ofspace and complexity. However,we believe that a discussionat
an elementary level of some aspects is surely part of a basic mathemati-
cal education and helps to understand the context in which the study of
abstract functions of many variables finds its true motivation.
Chapter1aimsatillustratinginconcretesituationstheabstracttreat-
ment of the geometry of Hilbert spaces that we presentedin [GM3]. After
a short illustration of Lebesgue’s spaces, in particular of L2, and a brief
introduction to Sobolev spaces, we present some complements to the the-
oryofFourierseries,themethodofseparationofvariablesfortheLaplace,
heatandwaveequations,andtheDirichletprincipleandweconcludewith
some results concerning the Sturm–Liouville theory. Chapter 2 is dedi-
catedtothetheoryofconvexfunctionsandtoillustratingseveralinstances
in which it naturally shows. Among these, the study of inequalities, the
Farkas lemma, and the linear and the convex programming with the the-
orem of Kuhn–Tucker and of von Neumann and Nash in the theory of
games. Chapter 3 is an introduction to calculus of variations. Our aim is
of just presenting the Lagrangian and Hamiltonian formalism, hinting at
some ofits connections withgeometricaloptics, mechanics,andsomegeo-
metrical examples. Chapter 4 deals with the general theory of differential
1 ThisbookisatranslationandarevisededitionofM.Giaquinta,G.ModicaAnalisi
Matematica, V. Funzioni di piu` variabili: ulteriori sviluppi, PitagoraEd.,Bologna,
2005.
2 M. Giaquinta, G. Modica, Mathematical Analysis, Functions of One Variable,
Birkhauser,Boston,2003,
M.Giaquinta,G.Modica,MathematicalAnalysis,ApproximationandDiscretePro-
cesses,Birkhauser,Boston, 2004,
M.Giaquinta,G.Modica,MathematicalAnalysis,LinearandMetricStructuresand
Continuity,Birkhauser,Boston,2007,
M.Giaquinta, G.Modica, Mathematical Analysis, An Introduction to Functions of
several variables,Birkhauser,Boston,2009.
Weshallrefertothesebooksasto[GM1],[GM2],[GM3]and[GM4],respectively.
v
vi Preface
formswiththeStokestheorem,thePoincar´elemma,andsomeapplications
of geometrical character. The final two chapters, 5 and 6, are dedicated
to the general theory of measure and integration, only outlined in [GM4],
andincludesthestudyofBorel,RadonandHausdorffmeasuresandofthe
theory of derivation of measures.
The study of this volume requires a strong effort compared to the one
requestedfor the firstfour volumes, both for the intrinsinc difficulties and
for the width and varieties of the topics that appear. On the other hand,
we believe that it is very useful for the reader to have a wide spectrum
of contexts in which the ideas have developed and play an important role
and some reasons for an analysis of the formal and structural foundations
that at first sight might appear excessive. However, we have tried to keep
a simple style of presentation, always providing examples, enlightening
remarks and exercises at the end of each chapter. The illustrations and
the bibliographical note provide suggestions for further readings.
We are greatly indebted to Cecilia Conti for her help in polishing our
first draft and we warmly thank her. We would also like to thank Paolo
Acquistapace, Timoteo Carletti, Giulio Ciraolo, Roberto Conti, Giovanni
Cupini, Matteo Focardi, Pietro Majer and Stefano Marmi for their com-
ments and their invaluable help in catching errorsand misprints, and Ste-
fan Hildebrandt for his comments and suggestions concerning especially
the choice of illustrations. Our special thanks also go to all members of
the editorial technical staff of Birkh¨auser for the excellent quality of their
work and especially to Katherine Ghezzi and the executive editor Ann
Kostant.
Note: We have tried to avoid misprints and errors. However, as most
authors, we are imperfect. We will be very grateful to anybody who is
willing to point out errors or misprints or wants to express criticism or
comments. Our e-mail addresses are
[e-mail] [e-mail]
We will try to keep up an errata corrige at the following webpages:
http://www.sns.it/~giaquinta
http://www.dma.unifi.it/~modica
http://www.dsi.unifi.it/~modica
Mariano Giaquinta
Giuseppe Modica
Pisa and Firenze
March 2011
Contents
1. Spaces of Summable Functions and Partial Differential
Equations............................................... 1
1.1 Fourier Series and Partial Differential Equations.......... 1
1.1.1 The Laplace, Heat and Wave Equations............ 1
a. Laplace’s and Poisson’s equation ............... 1
b. The heat equation ............................ 3
c. The wave equation ........................... 6
1.1.2 The method of separation of variables ............. 7
a. Laplace’s equation in a rectangle ............... 8
b. Laplace’s equation on a disk ................... 11
c. The heat equation ............................ 14
d. The wave equation ........................... 15
1.2 Lebesgue’s Spaces .................................... 16
1.2.1 The space L∞ .................................. 16
1.2.2 Lp spaces, 1≤p<+∞ .......................... 18
a. The Lp norm ................................ 18
b. Approximation............................... 20
c. Separability.................................. 22
d. Duality ..................................... 22
e. L2 is a separable Hilbert space ................. 23
f. Means ...................................... 23
1.2.3 Trigonometric series in L2........................ 25
1.2.4 The Fourier transform ........................... 28
a. The Fourier transform in S(Rn) ................ 29
b. The Fourier transform in L2 ................... 31
1.3 Sobolev Spaces....................................... 33
a. Strong derivatives ............................ 33
b. Weak derivatives ............................. 35
c. Absolutely continuous functions ................ 37
d. H1-periodic functions ......................... 37
e. Poincar´e’sinequality.......................... 40
f. Rellich’s compactness theorem ................. 41
g. Traces ...................................... 43
1.4 Existence Theorems for PDE’s ......................... 43
1.4.1 Dirichlet’s principle ............................. 43
a. The weak form of the equilibrium equation ...... 44
vii
viii Contents
b. The space H−1............................... 46
c. The abstract Dirichlet principle ................ 47
d. The Dirichlet problem......................... 48
e. Neumann problem............................ 51
f. Cauchy–Riemann equations.................... 54
1.4.2 The alternative theorem ......................... 54
1.4.3 The Sturm–Liouville theory ...................... 57
1.4.4 Convex functionals on H1........................ 63
0
1.5 Exercises ............................................ 63
2. Convex Sets and Convex Functions ..................... 67
2.1 Convex Sets ......................................... 67
a. Definitions................................... 67
b. The support hyperplanes ...................... 69
c. Convex hull.................................. 72
d. The distance function from a convex set......... 73
e. Extreme points............................... 76
2.2 Proper Convex Functions.............................. 76
a. Definitions................................... 76
b. A few characterizations of convexity ............ 77
c. Support function ............................. 78
d. Convex functions of class C1 and C2 ............ 80
e. Lipschitz continuity of convex functions ......... 82
f. Supporting planes and differentiability .......... 83
g. Extremal points of convex functions ............ 86
2.3 Convex Duality ...................................... 87
a. The polar set of a convex set................... 87
b. The Legendre transform for functions of
one variable ................................. 89
c. The Legendre transform for functions of
several variables.............................. 90
2.4 Convexity at Work ................................... 91
2.4.1 Inequalities .................................... 91
a. Jensen inequality ............................. 91
b. Inequalities for functions of matrices ............ 93
c. Doubly stochastic matrices .................... 94
2.4.2 Dynamics: Action and energy..................... 97
2.4.3 The thermodynamic equilibrium .................. 99
a. Pure and mixed phases........................102
2.4.4 Polyhedral sets .................................104
a. Regular polyhedra............................104
b. Implicit convex cones .........................105
c. Parametrized convex cones ....................106
d. Farkas–Minkowski’s lemma ....................108
2.4.5 Convex optimization ............................109
2.4.6 Stationary states for discrete-time Markov processes.112
2.4.7 Linear programming.............................114
Contents ix
a. The primal and dual problem ..................117
2.4.8 Minimax theorems and the theory of games ........121
a. The minimax theorem of von Neumann .........122
b. Optimal mixed strategies ......................127
c. Nash equilibria...............................128
d. Convex duality...............................130
2.5 A General Approach to Convexity ......................133
a. Definitions...................................133
b. Lower semicontinuous functions and closed
epigraphs....................................134
c. The Fenchel transform ........................138
d. Convex duality revisited.......................140
2.6 Exercises ............................................146
3. The Formalism of the Calculus of Variations............149
3.1 LagrangianFormalism ................................151
3.1.1 Euler–Lagrangeequations........................151
a. Dirichlet’s problem ...........................152
b. Natural boundary conditions...................154
c. Examples....................................155
3.1.2 Some remarks on the existence and regularity of
minimizers .....................................164
a. Existence....................................165
b. Regularity in the 1-dimensional case ............167
3.1.3 Constrained variational problems .................169
a. Isoperimetric constraints ......................170
b. Holonomic constraints.........................172
3.1.4 Noether’s theorem ..............................177
a. General variations ............................178
b. Inner variations ..............................179
c. Curves of minimal energy and curves of minimal
length.......................................181
d. Surfaces of minimal energy and surfaces of
minimal area.................................183
e. Noether theorem .............................184
3.1.5 The eikonal and the Huygens principle.............186
a. Calibrations and fields of extremals .............187
b. Mayer fields .................................189
c. The Weierstrass representation formula..........190
d. Huygens principle ............................191
3.2 The Classical Hamiltonian Formalism ...................193
3.2.1 The canonical equations of Hamilton and
Hamilton–Jacobi................................193
a. Hamilton equations...........................194
b. Liouville’s theorem ...........................195
c. Hamilton–Jacobi equation .....................195
d. Poincar´e–Cartanintegral ......................196