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Optimal convex shapes for concave functionals

Lamboley, Jimmy; Fragalà, Ilaria; Bucur, Dorin (2012), Optimal convex shapes for concave functionals, ESAIM. COCV, 18, 3, p. 693-711. http://dx.doi.org/10.1051/cocv/2011167

Type
Article accepté pour publication ou publié
External document link
http://hal.archives-ouvertes.fr/hal-00564691/fr/
Date
2012
Journal name
ESAIM. COCV
Volume
18
Number
3
Publisher
EDP Sciences
Pages
693-711
Publication identifier
http://dx.doi.org/10.1051/cocv/2011167
Metadata
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Author(s)
Lamboley, Jimmy
Fragalà, Ilaria
Bucur, Dorin
Abstract (EN)
Motivated by a long-standing conjecture of Polya and Szegö about the Newtonian capacity of convex bodies, we discuss the role of concavity inequalities in shape optimization, and we provide several counterexamples to the Blaschke-concavity of variational functionals, including capacity. We then introduce a new algebraic structure on convex bodies, which allows to obtain global concavity and indecomposability results, and we discuss their application to isoperimetriclike inequalities. As a byproduct of this approach we also obtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class of functionals involving Dirichlet energies and the surface measure, we perform a local analysis of strictly convex portions of the boundary via second order shape derivatives. This allows in particular to exclude the presence of smooth regions with positive Gauss curvature in an optimal shape for Polya-Szegö problem.
Subjects / Keywords
capacity; shape derivatives; optimization; concavity inequalities; convex bodies

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