Postmodern Analysis

364 Pages·2003·17.382 MB·English

by  Jürgen Jost (auth.)

Universitext
Springer-Verlag Berlin Heidelberg GmbH
Jiirgen Jost
Postmodern
Analysis
Second Edition
Springer
lilrgenlost
MaxPlanck Institutefor Mathematicsinthe Sciences
Inselstrasse 22-26
04103Leipzig
Germany
Cataloging-in-PublicationDataappliedfor
DieDeutscheBibliothek-CIP-Einheitsaufnahme
Jost.JOrgen:
PostmodernanalysisIJOrgenJost.-Berlin;Heidelberg;NewYork;
Barcelona;HongKong;London;Milan;Paris;Tokyo:Springer.2002
(Univenitext)
MathematicsSubjectOassification(2000):26-01,46-01,49-01
ISBN 978-3-540-43873-1 ISBN 978-3-662-05306-5 (eBook)
DOI 10.1 007/978-3-662-05306-5
ThisworkissubjecttocopyrightAllrightsarereserved,whetherthewholeorpartofthematerialis
concerned,specifically the rights of translation,reprinting.reuse of illustrations, recitation,
broadcasting.reproductiononmia061morinanyother-r.andstorageindatabankJ.Duplicationof
thispublicationorpartsthereofispmnittedonlyundertheprovisionsoftheGermanCopyrightLaw
ofSeptanber9.196S,initscurrentversion,andpennisaionforwemustal-rsbeobtainedfromSprin
ger-Verlag.ViolationsareIiableforprosecutionundertheGermanCopyrightLaw.
httyJIwww.apringer.de
oSpringer-VerlagBerlinHeidelberg1998,2003
OriginallypublishedbySpringer-VerlagBerlinHeidelbergNewYorkin2003.
Softcoverreprintofthehardcover2ndedition2003
Theweofgeneraldesaiptivenames,registerednames,trademarksetc.inthispublicationdoesnot
imply.evenintheabsenceofaspecificstatement,thatsuchnamesareexemptfromtherelevant
protectivelawsandregulationsandthereforefreeforgeneralwe.
Coverdesign:designlIrproduction,Heidelberg
'fYpesettingbytheauthorusinga¥ macropackage
Printedonacid-freepaper SPIN10878560 4113142db-S43210
To Eberhard Zeidler,
for building up our Institute with so
much enthusiasm and human skill
Preface
What is the title of this book intended to signify, what connotations is the
adjective "Postmodern" meant to carry? A potential reader will surely pose
this question. To answer it, I should describe what distinguishes the approach
to analysis presented here from what has been called "Modern Analysis" by
its protagonists. "Modern Analysis" as represented in the works of the Bour
baki group or in the textbooks by Jean Dieudonne is characterized by its
systematic and axiomatic treatment and by its drive towards a high level of
abstraction. Given the tendency of many prior treatises on analysis to degen
erate into a collection of rather unconnected tricks to solve special problems,
this definitely represented a healthy achievement. In any case, for the de
velopment of a consistent and powerful mathematical theory, it seems to be
necessary to concentrate solely on the internal problems and structures and
to neglect the relations to other fields of scientific, even of mathematical study
for a certain while. Almost complete isolation may be required to reach the
level of intellectual elegance and perfection that only a good mathematical
theory can acquire. However, once this level has been reached, it might be
useful to open one's eyes again to the inspiration coming from concrete ex
ternal problems. The axiomatic approach started by Hilbert and taken up
and perfected by the Bourbaki group has led to some of the most important
mathematical contributions of our century, most notably in the area of al
gebraic geometry. This development was definitely beneficial for many areas
of mathematics, but for other fields this was not true to the same extent. In
geometry, the powerful tool of visual imagination was somewhat neglected,
and global nonlinear phenomena connected with curvature could not always
be adequately addressed. In analysis, likewise, the emphasis was put perhaps
too much on the linear theory, while the genuinely nonlinear problems were
found to be too diverse to be subjected to a systematic and encompassing
theory. This effect was particularly noticeble in the field of partial differential
equations. This branch of mathematics is one of those that have experienced
the most active and mutually stimulating interaction with the sciences, and
those equations that arise in scientific applications typically exhibit some
genuinely nonlinear structure because of self-interactions and other effects.
Thus, modern mathematics has been concerned with its own internal
structure, and it has achieved great successes there, but perhaps it has lost
VIII Preface
a little of the stimulation that a closer interaction with the sciences can of
fer. This trend has been reversed somewhat in more recent years, and in
particular rather close ties have been formed again between certain areas of
mathematics and theoretical physics. Also, in mathematical research, the em
phasis has perhaps shifted a bit from general theories back to more concrete
problems that require more individual methods.
I therefore felt that it would be appropriate to present an introduction to
advanced analysis that preserves the definite achievements of the theory that
calls itself "modern" , but at the same time transcends the latter's limitations.
For centuries, "modern" in the arts and the sciences has always meant
"new" , "different from the ancient" , some times even "revolutionary", and so
it was an epithet that was constantly shifting from one school to its successor,
and it never stuck with any artistic style or paradigm of research. That only
changed in our century, when abstract functionality was carried to its extreme
in architecture and other arts. Consequently, in a certain sense, any new
theory or direction could not advance any further in that direction, but had
to take some steps back and take up some of the achievements of "premodern"
theories. Thus, the denomination "modern" became attached to a particular
style and the next generation had to call itself "postmodern". As argued
above, the situation in mathematics is in certain regards comparable to that,
and it thus seems logical to call "postmodern" an approach that tries to build
upon the insights of the modern theory, but at the same time wishes to take
back the latter's exaggerations.
Of course, the word "postmodern" does not at present have an altogether
positive meaning as it carries some connotations of an arbitrary and unprin
cipled mixture of styles. Let me assure the potential reader that this is not
intended by the title of the present book. I rather wish to give a coherent
introduction to advanced analysis without abstractions for their own sake
that builds a solid basis for the areas of partial differential equations, the
calculus of variations, functional analysis and other fields of analysis, as well
as for their applications to analytical problems in the sciences, in particular
the ones involving nonlinear effects.
Of course, calculus is basic for all of analysis, but more to the point, there
seem to be three key theories that mathematical analysis has developed in
our century, namely the concept of Banach space, the Lebesgue integral, and
the notion of abstract differentiable manifold. Of those three, the first two
are treated in the present book, while the third one, although closest to the
author's own research interests, has to wait for another book (this is not
quite true, as I did treat that topic in more advanced books, in particular in
"Riemannian Geometry and Geometric Analysis", Springer, 1995).
The Lebesgue integration theory joins forces with the concept of Banach
spaces when the LP and Sobolev spaces are introduced, and these spaces are
basic tools for the theory of partial differential equations and the calculus of
variations. (In fact, this is the decisive advantage of the Lebesgue integral
Preface IX
over the older notion, the so-called Riemann integral, that it allows the con
struction of complete normed function spaces, i.e. Hilbert or Banach spaces,
namely the LP and Sobolev spaces.) This is the topic that the book will lead
the reader to.
The organization of the book is guided by pedagogical principles. After
all, it originated in a course I taught to students in Bochum at the beginning
level of a specialized mathematics education. Thus, after carefully collect
ing the prerequisites about the properties of the real numbers, we start with
continuous functions and calculus for functions of one variable. The intro
duction of Banach spaces is motivated by questions about the convergence
of sequences of continuous or differentiable functions. We then develop some
notions about metric spaces, and the concept of compactness receives partic
ular attention. Also, after the discussion of the one-dimensional theory, we
hope that the reader is sufficiently prepared and motivated to be exposed
to the more general treatment of calculus in Banach spaces. After present
ing some rather abstract results, the discussion becomes more concrete again
with calculus in Euclidean spaces. The implicit function theorem and the
Picard-Linde16f theorem on the existence and uniqueness of solutions of or
dinary differential equations (ODEs) are both derived from the Banach fixed
point theorem.
In the second part, we develop the theory of the Lebesgue integral in Eu
clidean spaces. As already mentioned, we then introduce LP and Sobolev
spaces and give an introduction to elliptic partial differential equations
(PDEs) and the calculus of variations. Along the way, we shall see several
examples arising from physics.
In the table of contents, I have described the key notions and results of
each section, and so the interested reader can find more detailed information
about the contents of the book there.
This book presents an intermediate analysis course. Thus, its level is some
what higher than the typical introductory courses in the German university
system. Nevertheless, in particular in the beginning, the choice and presen
tation of material are influenced by the requirement of such courses, and
I have utilized some corresponding German textbooks, namely the analysis
o.
courses of Forster (Analysis I - III, Vieweg 1976ff.) and H. Heuser (Anal
ysis I, II, Teubner, 1980ff.). Although the style and contents of the present
book are dictated much more by pedagogical principles than it is the case
in J. Dieudonne's treatise Modern Analysis, Academic Press, 1960ff., there
is some overlap of content. Although typically the perspective of my book is
different, the undeniable elegance of reasoning that can be found in the trea
tise of Dieudonne nevertheless induced me sometimes to adapt some of his
arguments, in line with my general principle of preserving the achievements
of the theory that called itself modern.
For the treatment of Sobolev spaces and the regularity of solutions of ellip
tic partial differential equations, I have used D. Gilbarg, N. Trudinger, Elliptic
J( I>reface
Partial Differential Equations of Second Order, Springer, 21983, although of
course the presentation here is more elementary than in that monograph.
In checking the manuscript for this book, suggesting corrections and im
provements, and proofreading, I received the competent and dedicated help
of Felicia Bernatzki, Christian Gawron, Lutz Habermann, Xiaowei Peng,
Monika Reimpell, Wilderich Tuschmann, and Tilmann Wurzbacher. My orig
inal German text was translated by Hassan Azad into English. The typing
and retyping of several versions of my manuscript was performed with pa
tience and skill by Isolde Gottschlich. The figures were created by Micaela
Krieger with the aid of Harald Wenk and Ralf Muno. I thank them all for
their help without which this book would not have become possible.
Preface to the 2nd edition
For this edition, I have added some material on the qualitative behavior of
solutions of ordinary differential equations, some further details on LP and
Sobolev functions, partitions of unity, and a brief introduction to abstract
measure theory. I have also used this opportunity to correct some misprints
and two errors from the first edition. I am grateful to Horst Lange, C. G.
Simader, and Matthias Stark for pertinent comments. I also should like to
thank Antje Vandenberg for her excellent 1EXwork.
Leipzig, May 2002 Jurgen Jost