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Asymptotic analysis of shape functionals. (English) Zbl 1031.35020
Shape analysis goes back to early 1970-s, when sensitivity of fracture dependent on shape was studied by Novozhilov and others, who introduced the idea of shape functionals. Banichuk and the Danish school (Bensoe, Olhoff,…), J. P. Zolesio, and his coauthors, studied sensitivity with respect to changes of shape, making the shape of the boundary depend on a “small” parameter. Here a crucial problem arises of sensitivity of functionals to small perturbations of the boundary, when the boundaries are singularly perturbed. The domain contains small cavities, contributing to the complexity of the optimization problem. The present authors expand ideas considering asymptotic expansions in a topological space, in particular of matched asymptotic expansions. The novel feature of a series of papers of the authors and their associates is their use of the concept of topological derivative which in the understanding of the reviewer is a variation, or perhaps a generalization of the Gâteaux derivative.
Several ideas of this rather lengthy paper are continuation of a series of many previous papers of the authors. The past coauthors in papers on related topics include I. S. Zorin, A. Zochowski, W. G. Mazya, J. P. Zolesio.

MSC:
35B27 Homogenization in context of PDEs; PDEs in media with periodic structure
35C20 Asymptotic expansions of solutions to PDEs
35B25 Singular perturbations in context of PDEs
74P15 Topological methods for optimization problems in solid mechanics
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