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Functional Analysis Sobolev Spaces And Partial Differential Equations(Brezis)英文版,数分好教材,知乎多人荐之
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1 C
Haim Brezis
Functional Analysis,
Sobolev Spaces and Partial
Differential Equations
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Haim Brezis
Distinguished Professor
Department of Mathematics
Rutgers University
Piscataway, NJ 08854
USA
brezis@math.rutgers.edu
and
Professeur émérite, Université Pierre et Marie Curie (Paris 6)
and
Visiting Distinguished Professor at the Technion
Editorial board:
Sheldon Axler, San Francisco State University
Vincenzo Capasso, Università degli Studi di Milano
Carles Casacuberta, Universitat de Barcelona
Angus MacIntyre, Queen Mary, University of London
Kenneth Ribet, University of California, Berkeley
Claude Sabbah, CNRS, École Polytechnique
Endre Süli, University of Oxford
Wojbor Woyczyński, Case Western Reserve University
ISBN 978-0-387-70913-0 e-ISBN 978-0-387-70914-7
DOI 10.1007/978-0-387-70914-7
Springer New York Dordrecht Heidelberg London
Library of Congress Control Number: 2010938382
Mathematics Subject Classification (2010): 35Rxx, 46Sxx, 47Sxx
© Springer Science+Business Media, LLC 2011
All rights reserved. This work may not be translated or copied in whole or in part without the written
permission of the publisher (Springer Science+Business Media, LLC, 233 Spring Street, New York, NY
10013, USA), except for brief excerpts in connection with reviews or scholarly analysis. Use in connec-
tion with any form of information storage and retrieval, electronic adaptation, computer software, or by
similar or dissimilar methodology now known or hereafter developed is forbidden.
The use in this publication of trade names, trademarks, service marks, and similar terms, even if they are
not identified as such, is not to be taken as an expression of opinion as to whether or not they are subject
to proprietary rights.
Printed on acid-free paper
Springer is part of Springer Science+Business Media (www.springer.com)
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To Felix Browder, a mentor and close friend,
who taught me to enjoy PDEs through the
eyes of a functional analyst
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Preface
This book has its roots in a course I taught for many years at the University of
Paris. It is intended for students who have a good background in real analysis (as
expounded, for instance, in the textbooks of G. B. Folland [2], A. W. Knapp [1],
and H. L. Royden [1]). I conceived a program mixing elements from two distinct
“worlds”: functional analysis (FA) and partial differential equations (PDEs). The first
part deals with abstract results in FA and operator theory. The second part concerns
the study of spaces of functions (of one or more real variables) having specific
differentiability properties: the celebrated Sobolev spaces, which lie at the heart of
the modern theory of PDEs. I show how the abstract results from FA can be applied
to solve PDEs. The Sobolev spaces occur in a wide range of questions, in both pure
and applied mathematics. They appear in linear and nonlinear PDEs that arise, for
example, in differential geometry, harmonic analysis, engineering, mechanics, and
physics. They belong to the toolbox of any graduate student in analysis.
Unfortunately, FA and PDEs are often taught in separate courses, even though
they are intimately connected. Many questions tackled in FA originated in PDEs (for
a historical perspective, see, e.g., J. Dieudonné [1] and H. Brezis–F. Browder [1]).
There is an abundance of books (even voluminous treatises) devoted to FA. There
are also numerous textbooks dealing with PDEs. However, a synthetic presentation
intended for graduate students is rare. and I have tried to fill this gap. Students who
are often fascinated by the most abstract constructions in mathematics are usually
attracted by the elegance of FA. On the other hand, they are repelled by the never-
ending PDE formulas with their countless subscripts. I have attempted to present
a “smooth” transition from FA to PDEs by analyzing first the simple case of one-
dimensional PDEs (i.e., ODEs—ordinary differential equations), which looks much
more manageable to the beginner. In this approach, I expound techniques that are
possibly too sophisticated for ODEs, but which later become the cornerstones of
the PDE theory. This layout makes it much easier for students to tackle elaborate
higher-dimensional PDEs afterward.
A previous version of this book, originally published in 1983 in French and fol-
lowed by numerous translations, became very popular worldwide, and was adopted
as a textbook in many European universities. A deficiency of the French text was the
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