##
**Functional analysis, Sobolev spaces and partial differential equations.**
*(English)*
Zbl 1220.46002

Universitext. New York, NY: Springer (ISBN 978-0-387-70913-0/pbk; 978-0-387-70914-7/ebook). xiii, 599 p. (2011).

Functional analysis evolved as the natural gathering point for a number of different investigations into the solvability of many classes of problems, including differential or 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 French [H. Brezis, “Analyse fonctionnelle. Théorie et applications” (Collection Mathématiques Appliquées pour la Maîtrise; Paris: Masson) (1983; Zbl 0511.46001)] 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.

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 which have been treated and by the abstract results which are developed.

The content of the book is divided into 11 chapters, as follows: 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. Characterization 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. Following Chapter XI, there are solutions for selected exercises (a significant number of complete solutions for the exercises following each chapter are included). Next, a list of 51 problems follows. 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.

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 a certain applicability to the real world. 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 emphasized 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, and for independent study.

A previous version of this book, originally published in 1983 in French [H. Brezis, “Analyse fonctionnelle. Théorie et applications” (Collection Mathématiques Appliquées pour la Maîtrise; Paris: Masson) (1983; Zbl 0511.46001)] 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.

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 which have been treated and by the abstract results which are developed.

The content of the book is divided into 11 chapters, as follows: 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. Characterization 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. Following Chapter XI, there are solutions for selected exercises (a significant number of complete solutions for the exercises following each chapter are included). Next, a list of 51 problems follows. 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.

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 a certain applicability to the real world. 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 emphasized 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, and for independent study.

Reviewer: Vicenţiu D. Rădulescu (Craiova)

### MathOverflow Questions:

An alternate definition of Sobolev space \(W^{1,p}(\Omega)\) when \(1<p\leq\infty\) and consequences### MSC:

46-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to functional analysis |

47-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to operator theory |

35-01 | Introductory exposition (textbooks, tutorial papers, etc.) pertaining to partial differential equations |

46E35 | Sobolev spaces and other spaces of “smooth” functions, embedding theorems, trace theorems |

46N20 | Applications of functional analysis to differential and integral equations |

47F05 | General theory of partial differential operators |