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Shape derivatives and differentiability of min max. (English) Zbl 0780.49029
Shape optimization and free boundaries, Proc. NATO ASI, Sémin. Math. Supér., Montréal/Can. 1990, NATO ASI Ser., Ser. C 380, 35-111 (1992).
[For the entire collection see Zbl 0753.00022.]
This paper deals with different questions related to the first and second-order analysis of a domain functional \(\Omega\in A\mapsto J(\Omega)\in{\mathcal R}\). Here \(A\) is a family of subsets (called admissible domains) of a given domain \(D\subset{\mathcal R}^ n\). Due to the lack of linear structure in \(A\), first and second-order derivatives of domain functionals have to be understood in a generalized sense. The author discusses several types of derivatives and develops some calculus rules. The last three sections are devoted to the case in which the domain functional \(J\) admits a Min formulation, or a Min-Max formulation. Some results have been presented in earlier works by the author and J. P. Zolésio [see, e.g., SIAM J. Control Optimization 26, No. 4, 834-862 (1988; Zbl 0654.49010) and J. Funct. Anal. 104, 1-33 (1992)].
Reviewer: A.Seeger (Dhahran)

49Q10 Optimization of shapes other than minimal surfaces
49J50 Fréchet and Gateaux differentiability in optimization
49J52 Nonsmooth analysis