1.1. From Contents.

CHAPTER I. SOME PRELIMINARY RESULTS
1. Min-Max Theorem.
2. Fixed Points of Contracting Maps.
3. Ordinary Differential Equations on Convex Sets.

CHAPTER II. MAXIMUM MONOTONE OPERATORS
1. Concept of monotone operators.
2. Concept of maximal monotone operators.
3. Examples of maximal monotone operators.
4. Elementary properties of maximal monotone operators.
5. Surjectivity of maximal monotone operators.
6. Sum of maximal monotone operators.
7. Cyclically monotone operators.
8. Examples of cyclically monotone operators.
9. Cyclically monotone perturbations.

CHAPTER III. EVOLUTION EQUATIONS ASSOCIATED WITH MONOTONE OPERATORS

CHAPTER IV. PROPERTIES OF NONLINEAR CONTRACTION SEMI-GROUPS
1. A nonlinear version of the Hille-Yosida-Phillips theorem.
2. Convergence properties: Neveu-Trotter-Kato theorem for nonlinear semi-groups.
3. Approximation of non-linear semi-groups: exponential formula, Chernoff and Trotter formulas.
4. Invariant subsets: convex Lyapunov functions and monotone operators.

1.2. Review by: Bruce Calvert.
Mathematical Reviews MR0348562 (50 #1060).

This excellent book deals with maximal monotone operators and semigroups of contractions in Hilbert space. ...
...
The material covered in this book has been developed in the last decade. Many significant results are due to the author. He has given a very clearly written and carefully developed treatment of this important theory. New material is included, as well as a thorough reworking of standard results. The treatment is essentially self-contained and up to date. This book is a must for researchers and graduate students interested in nonlinear operator.
...
The book is based on a course on nonlinear analysis given in Paris in 1970/71.

2.1. From the Introduction.

This text, which is based on a Master's course taught at the Pierre and Marie Curie University (Paris VI), assumes knowledge of the basic elements of general topology, integration and differential calculus. The first part of the course (Chapters I to VII) develops 'abstract' results of functional analysis. The second part (Chapters VIII to X) concerns the study of 'concrete' function spaces, which occur in the theory of partial differential equations; we show how 'abstract' existence theorems can be used to solve partial differential equations. These two branches of analysis are closely linked. Historically, 'abstract' functional analysis was developed to answer questions raised by the solution of partial differential equations. Conversely, progress in 'abstract' functional analysis has greatly stimulated the theory of partial differential equations. We omit historical references; the reader should consult J Dieudonne's History of functional analysis. We hope that this book will appeal to students of pure mathematics as well as those interested in applied mathematics.

I thank the Mathematics Research Center, University of Wisconsin, and the Department of Mathematics, University of Chicago, where parts of this book were written.

I dedicate this book to the memory of Guido Stampacchia, in homage to a Master of Functional Analysis, who passed away prematurely.

2.2. From the Contents.

I. The Hahn-Banach theorems. Introduction to the theory of conjugate convex functions;
II. The Banach-Steinhaus and closed graph theorems. Orthogonality relations. Unbounded operators. The notion of adjoint. Characterisation of surjective operators;
III. Weak topologies. Reflexive spaces. Separable spaces. Uniformly convex spaces;
IV. $L^{p}$-spaces;
V. Hilbert spaces;
VI. Compact operators. Spectral decomposition of compact selfadjoint operators;
VII. The Hille-Yosida theorem;
VIII. Sobolev spaces and variational formulation of boundary value problems in dimension one;
IX. Sobolev spaces and variational formulation of boundary value problems in dimension N;
X. Evolution problems: the heat equation and the wave equation.

3.1. From the Publisher.

This book is concerned with the study in two dimensions of stationary solutions of $u_{\epsilon}$ of a complex valued Ginzburg-Landau equation involving a small parameter ε. Such problems are related to questions occurring in physics, e.g., phase transition phenomena in superconductors and superfluids. The parameter ε has a dimension of a length which is usually small. Thus, it is of great interest to study the asymptotics as ε tends to zero.

One of the main results asserts that the limit u-star of minimisers $u_{\epsilon}$ exists. Moreover, u-star is smooth except at a finite number of points called defects or vortices in physics. The number of these defects is exactly the Brouwer degree - or winding number - of the boundary condition. Each singularity has degree one - or as physicists would say, vortices are quantised.

The material presented in this book covers mostly original results by the authors. It assumes a moderate knowledge of nonlinear functional analysis, partial differential equations, and complex functions. This book is designed for researchers and graduate students alike, and can be used as a one-semester text. The present softcover reprint is designed to make this classic text available to a wider audience.

3.2. From the Acknowledgements.

Part of this work was done while the first author (F.B.) and the third author (F.H.) were visiting Rutgers University. They thank the Mathematics Department for its support and hospitality; their work was also partially supported by a Grant of the French Ministry of Research and Technology (MRT Grant 90S0315). Part of this work was done while the second author (H.B.) was visiting the Scuola Normale Superiore of Pisa; he is grateful to the Scuola for its invitation. We also thank Lisa Magretto and Barbara Miller for their enthusiastic and competent typing of the manuscript.

3.3. From the Contents.

Introduction
I. Energy estimates for $S^{1}$-valued maps
II. A lower bound for the energy of $S^{1}$-valued maps on perforated domains
III. Some basic estimates for $u_{\epsilon}$
IV. Towards locating the singularities: bad discs and good discs
V. An upper bound for the energy of $u_{\epsilon}$ away from the singularities
VI. $u_{\epsilon_{n}}$ converges: $u_{^{*}}$ is born!
VII. $u_{^{*}}$ coincides with THE canonical harmonic map having singularities $(a_{j})$
VIII. The configuration $(a_{j})$ minimises the renormalised energy $W$
IX. Some additional properties of $u_{\epsilon}$
X. Non minimising solutions of the Ginzburg-Landau equation
XI. Open problems

4.1. From the Publisher.

Uniquely, this book presents a coherent, concise and unified way of combining elements from two distinct "worlds," functional analysis (FA) and partial differential equations (PDEs), and is intended for students who have a good background in real analysis. This text presents a smooth transition from FA to PDEs by analysing in great detail the simple case of one-dimensional PDEs (i.e., ODEs), a more manageable approach for the beginner. Although there are many books on functional analysis and many on PDEs, this is the first to cover both of these closely connected topics. Moreover, the wealth of exercises and additional material presented, leads the reader to the frontier of research. This book has its roots in a celebrated course taught by the author for many years and is a completely revised, updated, and expanded English edition of the important "Analyse Fonctionnelle" (1983). Since the French book was first published, it has been translated into Spanish, Italian, Japanese, Korean, Romanian, Greek and Chinese. The English version is a welcome addition to this list. The first part of the text deals with abstract results in FA and operator theory. The second part is concerned with the study of spaces of functions (of one or more real variables) having specific differentiability properties, e.g., the celebrated Sobolev spaces, which lie at the heart of the modern theory of PDEs. The Sobolev spaces occur in a wide range of questions, both in pure and applied mathematics, appearing in linear and nonlinear PDEs which arise, for example, in differential geometry, harmonic analysis, engineering, mechanics, physics etc. and belong in the toolbox of any graduate student studying analysis.

4.2. From the 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, A W Knapp, and H L Royden. 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 and H Brezis-F Browder). 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 analysing 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 followed 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 lack of exercises. The present book contains a wealth of problems. I plan to add even more in future editions. I have also outlined some recent developments, especially in the direction of nonlinear PDEs.

4.3. From the Contents.

I. The Hahn-Banach theorems. Introduction to the theory of conjugate convex functions;
II. The Banach-Steinhaus and closed graph theorems. Orthogonality relations. Unbounded operators. The notion of adjoint. Characterisation of surjective operators;
III. Weak topologies. Reflexive spaces. Separable spaces. Uniformly convex spaces;
IV. $L^{p}$-spaces;
V. Hilbert spaces;
VI. Compact operators. Spectral decomposition of compact selfadjoint operators;
VII. The Hille-Yosida theorem;
VIII. Sobolev spaces and variational formulation of boundary value problems in dimension one;
IX. Sobolev spaces and variational formulation of boundary value problems in dimension N;
X. Evolution problems: the heat equation and the wave equation;
XI. Miscellaneous complements.
Solutions for selected exercises.
List of 51 problems.
Hints and partial solutions for the set of 51 problems.

4.4. Review by: Vicentiu D Radulescu.
Mathematical Reviews MR2759829 (2012a:35002).

Functional analysis evolved as a natural gathering point for a number of investigations regarding the solvability of many classes of problems, including ordinary and partial differential equations, or linear equations which are either in the form of integral equations or in the form of countable systems of linear scalar equations in which the unknown is a sequence of numbers. As the subject developed, much broader areas of applicability became evident. These applications, in turn, spawned further abstract development, and the abstract results themselves assumed an intrinsic interest.

A previous version of this book, originally published in 1983 in [H R Brezis, Analyse fonctionnelle, 1983] and followed 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 lack of exercises. The present volume contains a wealth of problems whose solution is left to the reader.

The author's aim is to give a systematic treatment of some of the fundamental abstract results in functional analysis and of their applications to certain concrete problems in linear differential and partial differential equations. Moreover, by interlacing extensive commentary and foreshadowing subsequent developments within the formal scheme of statements and proofs, by the inclusion of many apt examples and by appending interesting and challenging exercises, the author has written a book which is eminently suitable as a text for a graduate course. This volume is distinguished by the broad variety of problems covered and by the abstract results developed.
...
In summary, this book is a tour de force by the author, who is a master of modern nonlinear functional analysis and who has contributed extensively to the development of the theory of partial differential equations. The volume under review deals rigorously with mathematical models of certain applicability to real world problems. From this viewpoint it is a significant contribution to a currently active area of research.

Globally, the reviewer very much likes the spirit and the scope of the book. The writing is lively, the material is diverse and maintains a strong unity. The interplay between the abstract functional analysis and relevant concrete problems arising in applications is emphasised throughout. On balance, the book is a very useful contribution to the growing literature on this circle of ideas. I wholeheartedly recommend this book both as a textbook, as well as for independent study.

4.5. Review by: Florin Catrina.
Mathematical Association of America (28 April 2011).
https://web.archive.org/web/20151018141846/https://www.maa.org/publications/maa-reviews/functional-analysis-sobolev-spaces-and-partial-differential-equations

The author of this book, Haim Brezis, is one of the world's top researchers in the area of Partial Differential Equations. He is an excellent presenter, and, as the readers of this book will quickly realize, he is also a master communicator in writing.

The book is the English translation of an 1983 book published in French: Analyse fonctionnelle : théorie et applications, (Dunod, Paris, reprinted in 2005). It has seen translations into numerous languages and the Springer edition was especially anticipated, as it announced a number of practice exercises following each chapter. I can honestly say that it was well worth the wait.

The structure of the book is as follows: the first six chapters deal with Functional Analysis; chapters seven through ten introduce and then focus mostly on Partial Differential Equations; chapter eleven is a collection of various facts, mainly related to the first seven chapters. At the end of each chapter there are exercises, many of which contain hints, and are broken down into parts that lead progressively to the main result.

Following Chapter 11, there are solutions for selected exercises (a significant number of complete solutions for the exercises following each chapter are included). Next, follows a list of 51 Problems. These are mostly theorems, or otherwise theoretical results, the proofs of which could be treated separately from the main text. Finally, there is a section of hints and partial solutions for the set of 51 problems.

The text is a pleasure to read. There are numerous interesting remarks and comments spread throughout, and at the end of each chapter there is a subsection of comments. The comments point to interesting areas of research related to the topics discussed and to a wealth of references to current literature.

I wholeheartedly recommend this book as both a textbook as well as for independent study.

4.6. Review by: G Teschl.
Monatshefte für Mathematik 165 (3-4) (2012), 601-602.

This textbook has its origin in the French version Analyse fonctionnelle published in 1985, which has become a standard reference and was translated into several languages.

The present version continues this tradition and now combines two courses, one on functional analysis and one on applications to partial differential equations. Building on a solid basis in real analysis, the first part introduces the reader to basic concepts in functional analysis and operator theory. The usual topics including the Hahn-Banach theorem, the Baire category theorem and its consequences, weak topologies, Lebesgue Lp spaces, Hilbert spaces, and compact operators are discussed here. The second part deals with applications to partial differential equations. It starts out with operator semigroups and then discusses Sobolev spaces and the variational formulation of elliptic boundary value problems. What is unusual here, is the fact that the author introduces the concepts first in one dimension (i.e. for Sturm-Liouville problems) and then shows how they extend to higher dimensions. In particular, this will significantly lower the entry hurdle for beginners. Finally, there is material on the heat and wave equation. At the end of each chapter the reader will find comments with further information, references, and historic remarks. Moreover, there one will also find a large number of exercises (something which was missing in the French original) with hints (and some of them even with solutions). Moreover, in an appendix there are also somewhat more involved problems (again with partial solutions). In summary, the present textbook provides an excellent basis for a course on functional analysis plus a follow-up course on partial differential equations. It is well-written and I can wholeheartedly recommend it to both students and teachers.

5.1. From the Publisher.

The theory of real-valued Sobolev functions is a classical part of analysis and has a wide range of applications in pure and applied mathematics. By contrast, the study of manifold-valued Sobolev maps is relatively new. The incentive to explore these spaces arose in the last forty years from geometry and physics. This monograph is the first to provide a unified, comprehensive treatment of Sobolev maps to the circle, presenting numerous results obtained by the authors and others. Many surprising connections to other areas of mathematics are explored, including the Monge-Kantorovich theory in optimal transport, items in geometric measure theory, Fourier series, and non-local functionals occurring, for example, as denoising filters in image processing. Numerous digressions provide a glimpse of the theory of sphere-valued Sobolev maps.

Each chapter focuses on a single topic and starts with a detailed overview, followed by the most significant results, and rather complete proofs. The "Complements and Open Problems" sections provide short introductions to various subsequent developments or related topics, and suggest new directions of research. Historical perspectives and a comprehensive list of references close out each chapter. Topics covered include lifting, point and line singularities, minimal connections and minimal surfaces, uniqueness spaces, factorisation, density, Dirichlet problems, trace theory, and gap phenomena.

Sobolev Maps to the Circle will appeal to mathematicians working in various areas, such as nonlinear analysis, PDEs, geometric analysis, minimal surfaces, optimal transport, and topology. It will also be of interest to physicists working on liquid crystals and the Ginzburg-Landau theory of superconductors.

5.2. From the Contents.

(1) Lifting in $W^{\{1,p\}}$;
(2) The geometry of $Ju$ and $\Sigma (u)$ in 2D; point singularities and minimal connections;
(3) The geometry of $Ju$and $\Sigma (u)$ in 3D (and higher); line singularities and minimal surfaces;
(4) A digression: sphere-valued maps;
(5) Lifting in fractional Sobolev spaces and in VMO;
(6) Uniqueness of lifting and beyond;
(7) Factorisation;
(8) Application of the factorisation;
(9) Estimates of phases: positive and negative results;
(10) Density;
(11) Traces;
(12) Degree;
(13) Dirichlet problems. Gaps. Infinite energies;
(14) Domains with topology;
(15) Appendices.

5.3. Review by: Antonia Chinnì.
Mathematical Reviews MR4390036.

This excellent monograph is the result of travel through the "enchanted world of sphere-valued maps" undertaken by the authors with the intention of collecting "everything you wanted to know" on this topic. The work collects a series of contributions obtained by the authors from the 1980s to now. The extensive bibliography (349 titles) underlines the fruitful collaboration with authoritative mathematicians, many of whom have joined this intriguing journey following various courses, meetings and visits held by the authors in prestigious academic locations.

The book is divided into 15 chapters whose content is wisely summarised in an overview. Each chapter consists of an introduction that offers a detailed summary of the results that will be established in the various sections. The section entitled "Complements and open problems", at the end of each chapter, contains suggestions for new directions of research and indications on "what needs to be modified or what remains to be proved" when the target space is $S^{k}$.
...
In summary, this monograph offers a rigorous discussion to a fascinating topic of mathematics with multiple relevant applications to various fields. Also for this reason it is highly recommended both for mathematicians and physicists working in the various fields involved in the theory of Sobolev maps to the circle (nonlinear analysis, PDEs, geometric analysis, minimal surfaces, optimal transport, topology, liquid crystals, Ginzburg-Landau theory of superconductors). Moreover, the book may also be useful for Ph.D. students who are interested in the topic, and the content of the individual chapters could be used in advanced courses and seminars.