##
**Modern methods in the calculus of variations. \(L^p\) spaces.**
*(English)*
Zbl 1153.49001

Springer Monographs in Mathematics. New York, NY: Springer (ISBN 978-0-387-35784-3/hbk; 978-0-387-69006-3/ebook). xiv, 599 p. (2007).

This book is the first of two volumes in the calculus of variations and measure theory. The main objective of this book is to introduce necessary and sufficient conditions for sequential lower semicontinuity of functionals on \(L^p\)-spaces.

It is divided into three parts plus an appendix. The first part covers background material on measure theory, integration and \(L^p\)-spaces. It contains new proofs and some results are not restricted to the context of \(\sigma\)-finite measures. The authors have in mind the treatment of Hausdorff measures which will play an important role in volume 2. The second part is devoted to the direct method in the calculus of variations and to some results for convex functions and in convex analysis (regularity of convex functions, recession functions, convex envelopes…).

The third part is dedicated to functionals defined on \(L^p\)-spaces. The analysis of lower semicontinuity is performed for different topologies: strong convergence in \(L^p\), weak and weak star convergence in \(L^p\), weak star convergence in the sense of measures and weak star convergence in the dual of the space of bounded and continuous functions over an open subset of \({\mathbb R}^n\). Various types of functionals are studied, but not functionals depending on gradients which will be studied in volume 2. This part ends with an introduction to relaxation via Young measures.

Finally in the appendix some results in functional analysis, set theory, topological vector spaces are recalled. This book is very nicely written, self-contained and it is an excellent and modern introduction to the calculus of variations. I am waiting impatiently the second volume. Finally, I would like to mention that notes and a list of open problems are included at the end of the volume.

It is divided into three parts plus an appendix. The first part covers background material on measure theory, integration and \(L^p\)-spaces. It contains new proofs and some results are not restricted to the context of \(\sigma\)-finite measures. The authors have in mind the treatment of Hausdorff measures which will play an important role in volume 2. The second part is devoted to the direct method in the calculus of variations and to some results for convex functions and in convex analysis (regularity of convex functions, recession functions, convex envelopes…).

The third part is dedicated to functionals defined on \(L^p\)-spaces. The analysis of lower semicontinuity is performed for different topologies: strong convergence in \(L^p\), weak and weak star convergence in \(L^p\), weak star convergence in the sense of measures and weak star convergence in the dual of the space of bounded and continuous functions over an open subset of \({\mathbb R}^n\). Various types of functionals are studied, but not functionals depending on gradients which will be studied in volume 2. This part ends with an introduction to relaxation via Young measures.

Finally in the appendix some results in functional analysis, set theory, topological vector spaces are recalled. This book is very nicely written, self-contained and it is an excellent and modern introduction to the calculus of variations. I am waiting impatiently the second volume. Finally, I would like to mention that notes and a list of open problems are included at the end of the volume.

Reviewer: Jean-Pierre Raymond (Toulouse)

### MSC:

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

49J45 | Methods involving semicontinuity and convergence; relaxation |

28-02 | Research exposition (monographs, survey articles) pertaining to measure and integration |

28B20 | Set-valued set functions and measures; integration of set-valued functions; measurable selections |

52Axx | General convexity |