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Distributed and boundary expressions of first and second order shape derivatives in nonsmooth domains. (English. French summary) Zbl 1431.49050

Summary: We study distributed and boundary integral expressions of Eulerian and Fréchet shape derivatives for several classes of nonsmooth domains such as open sets, Lipschitz domains, polygons and curvilinear polygons, semiconvex and convex domains. For general shape functionals, we establish relations between distributed Eulerian and Fréchet shape derivatives in tensor form for Lipschitz domains, and infer two types of boundary expressions for Lipschitz and \(\mathcal{C}^1\)-domains. We then focus on the particular case of the Dirichlet energy, for which we compute first and second order distributed shape derivatives in tensor form. Depending on the type of nonsmooth domain, different boundary expressions can be derived from the distributed expressions. This requires a careful study of the regularity of the solution to the Dirichlet Laplacian in nonsmooth domains. These results are applied to obtain a matricial expression of the second order shape derivative for polygons.

MSC:

49Q10 Optimization of shapes other than minimal surfaces
49J52 Nonsmooth analysis
49Q12 Sensitivity analysis for optimization problems on manifolds
35Q93 PDEs in connection with control and optimization
35R37 Moving boundary problems for PDEs

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