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**Concentration compactness. Functional-analytic grounds and applications.**
*(English)*
Zbl 1118.49001

London: Imperial College Press (ISBN 978-1-86094-666-0/hbk; 978-1-86094-667-7/pbk). xii, 264 p. (2007).

From the authors’ preface: The subject of this book, concentration compactness, is a method for establishing convergence, in functional spaces, of sequences that are not a priori located in a compact set. This situation occurs, in particular, in variational problems with functionals that are invariant under some non-compact group of operators, and therefore have non-compact level sets. The concentration compactness argument considers possible “dislocated” limits of the sequence, that is, limits under sequences of the “gauge” operators. The proof of convergence then can be based on elimination of the dislocated limits. …

We have selected the contents for the book in order to give an accessible, rather than technical presentation for the concentration compactness. We have opted to present the topic in Hilbert space, rather that Banach space, and included four chapters with background material: chapter 1 – a compilation of theorems from functional analysis, chapter 2 – a compendium on Sobolev spaces with focus on \(H^1(\Omega)\) and unbounded sets, and chapters 7–8 on differentiable manifolds and Lie groups. The reader is expected to be familiar with basics of point-set topology, metric spaces, and measure theory. The presentation of Sobolev spaces in chapter 2 implicitly emphasizes the role of conformal group of Euclidean space, an approach which is later generalized in the concentration compactness argument for a conformal group of a manifold in the treatment of sub-elliptic Sobolev spaces in chapter 9. The functional-analytic grounds of the concentration compactness are presented in chapter 3, followed by applications in chapters 4, 5, and 6 to functions on Euclidean domains. Chapter 9 is an introduction of sub-elliptic Sobolev spaces on Lie groups, followed by some analogs of problems considered in the preceding chapters that involve sub-elliptic operators and “magnetic” Laplace-Beltrami operators on manifolds. The authors will use a follow-up web page http://www.math.uu.se/~tintarev/cc.html to provide additional materials, problems, corrections etc.

We have selected the contents for the book in order to give an accessible, rather than technical presentation for the concentration compactness. We have opted to present the topic in Hilbert space, rather that Banach space, and included four chapters with background material: chapter 1 – a compilation of theorems from functional analysis, chapter 2 – a compendium on Sobolev spaces with focus on \(H^1(\Omega)\) and unbounded sets, and chapters 7–8 on differentiable manifolds and Lie groups. The reader is expected to be familiar with basics of point-set topology, metric spaces, and measure theory. The presentation of Sobolev spaces in chapter 2 implicitly emphasizes the role of conformal group of Euclidean space, an approach which is later generalized in the concentration compactness argument for a conformal group of a manifold in the treatment of sub-elliptic Sobolev spaces in chapter 9. The functional-analytic grounds of the concentration compactness are presented in chapter 3, followed by applications in chapters 4, 5, and 6 to functions on Euclidean domains. Chapter 9 is an introduction of sub-elliptic Sobolev spaces on Lie groups, followed by some analogs of problems considered in the preceding chapters that involve sub-elliptic operators and “magnetic” Laplace-Beltrami operators on manifolds. The authors will use a follow-up web page http://www.math.uu.se/~tintarev/cc.html to provide additional materials, problems, corrections etc.

Reviewer: Corneliu Ursescu (Iaşi)

### MSC:

49-02 | Research exposition (monographs, survey articles) pertaining to calculus of variations and optimal control |

35-02 | Research exposition (monographs, survey articles) pertaining to partial differential equations |

46-02 | Research exposition (monographs, survey articles) pertaining to functional analysis |

58-02 | Research exposition (monographs, survey articles) pertaining to global analysis |