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Shapes and geometries. Analysis, differential calculus, and optimization. (English) Zbl 1002.49029
Advances in Design and Control. 4. Philadelphia, PA: SIAM. xvii, 482 p. (2001).
In many mathematical, physical and industrial applications, the geometry of the domain has great consequences on the behavior of the system. Many examples can be be found in the companion book “Introduction to shape optimization: shape sensitivity analysis” (1992; Zbl 0761.73003) by J. Sokolowski and J.-P. Zolésio.
The objective of Shape and geometries is to give an extended mathematical setting. General situations include non-smooth and large evolution domains. After the Introduction in Chapter 1, the book is divided in two parts. The first part, containing Chapters 2 to 6, deals with the analysis of domains and their properties, while the second part, including Chapters 7 to 9, pertains to the choice of perturbations of domains to define shape derivative.
The main areas being discussed are the Courant topology, the intrinsic geometric approach (including the oriented distance function), the compacity results, the equivalence between transformation and velocity, and the min-max differentiability, which avoids differentiation of the state equation. The non-smooth analysis allows adaptation of the technique to image analysis while the speed method is a generalization of the so-called level set approach. The oriented distance function presented in Chapter 5 has been successfully applied by the authors to shell modeling and will be fully developed in a book to appear. The authors chose not to cover some recent work concerning weak shape evolution associated with non-Lipschitzian vector field. Such vector fields allowing topological changes can be found, for instance, in “Shape optimization and optimal design” (1999; Zbl 0959.00036) edited by J. Cagnol, M. P. Polis and J.-P. Zolésio.
The book Shape and geometries is a “must have” for mathematics, physics and engineering libraries. Though the concepts covered are elaborate, the text is written very clearly and does not require more than the standard mathematical background material. It is an excellent reference and can be used as textbook for a graduate class. It should be mentioned that both authors are authorities in the area of shape optimization.

49Q10 Optimization of shapes other than minimal surfaces
49-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to calculus of variations and optimal control