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**Variation and optimization of formes. A geometric analysis.
(Variation et optimisation de formes. Une analyse géométrique.)**
*(French)*
Zbl 1098.49001

Mathématiques & Applications (Berlin) 48. Berlin: Springer (ISBN 3-540-26211-3/pbk). xii, 334 p. (2005).

The book under review is devoted to the very interesting field of shape optimization problems, which received a lot of attention in the recent years. The input for this kind of problems comes from different directions: in mathematics because of the elegance of handling shapes in several geometric optimization problems (as the isoperimetric one, the eigenvalues optimization, etc.) as well as in engineering, where a careful analysis on the shapes of mechanical pieces may lead to quit big improvements in their physical performances. The book does not neglect any of these aspects: from the one hand the mathematical tools are introduced with a highly rigorous approach, from the other hand many examples coming from applications are presented.

After a first introductory chapter containing several examples, an entire chapter is devoted to the presentation of the various topologies on families of domains of \(\mathbb{R}^N\), an essential tool for every shape optimization problem. Chapter 3 deals with the related analysis of the continuity properties of the solutions of a PDE under the modification of the corresponding domain. Chapter 4 uses the analysis of the previous chapters to attack the question of existence of optimal shapes and to discuss the oscillating behaviour of the minimizing sequences when this existence fails. Chapter 5 deals with the derivation with respect to a domain, which is crucial for establishing some necessary condition of optimality. In Chapter 6 some geometrical properties of optimal shapes are studied, and finally in Chapter 7 some links to other theories as relaxation, homogenization, \(\Gamma\)-convergence, are illustrated.

The book also includes many exercises which, together with all the remarks, comments and presentations of problems still open, make the volume an excellent choice for anyone who wants to approach the fascinating field of shape optimization.

After a first introductory chapter containing several examples, an entire chapter is devoted to the presentation of the various topologies on families of domains of \(\mathbb{R}^N\), an essential tool for every shape optimization problem. Chapter 3 deals with the related analysis of the continuity properties of the solutions of a PDE under the modification of the corresponding domain. Chapter 4 uses the analysis of the previous chapters to attack the question of existence of optimal shapes and to discuss the oscillating behaviour of the minimizing sequences when this existence fails. Chapter 5 deals with the derivation with respect to a domain, which is crucial for establishing some necessary condition of optimality. In Chapter 6 some geometrical properties of optimal shapes are studied, and finally in Chapter 7 some links to other theories as relaxation, homogenization, \(\Gamma\)-convergence, are illustrated.

The book also includes many exercises which, together with all the remarks, comments and presentations of problems still open, make the volume an excellent choice for anyone who wants to approach the fascinating field of shape optimization.

Reviewer: Giuseppe Buttazzo (Pisa)

### MSC:

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

74-02 | Research exposition (monographs, survey articles) pertaining to mechanics of deformable solids |

49Q10 | Optimization of shapes other than minimal surfaces |

49Q05 | Minimal surfaces and optimization |

49Q12 | Sensitivity analysis for optimization problems on manifolds |

49J45 | Methods involving semicontinuity and convergence; relaxation |

35R35 | Free boundary problems for PDEs |