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Optimal convex shapes for concave functionals

Published online by Cambridge University Press:  29 September 2011

Dorin Bucur ,
Ilaria Fragalà  and
Jimmy Lamboley
Dorin Bucur
Affiliation:
Laboratoire de Mathématiques UMR 5127, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-du-Lac, France
Ilaria Fragalà
Affiliation:
Dipartimento di Matematica, Politecnico, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy. ilaria.fragala@polimi.it
Jimmy Lamboley
Affiliation:
Ceremade UMR 7534, Université de Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France
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Abstract

Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacityof convex bodies, we discuss the role of concavity inequalities in shape optimization, andwe provide several counterexamples to the Blaschke-concavity of variational functionals,including capacity. We then introduce a new algebraic structure on convex bodies, whichallows to obtain global concavity and indecomposability results, and we discuss theirapplication to isoperimetric-like inequalities. As a byproduct of this approach we alsoobtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class offunctionals involving Dirichlet energies and the surface measure, we perform a localanalysis of strictly convex portions of the boundary via second ordershape derivatives. This allows in particular to exclude the presence of smooth regionswith positive Gauss curvature in an optimal shape for Pólya-Szegö problem.

Keywords

Convex bodiesconcavity inequalitiesoptimizationshape derivativescapacity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 693 - 711
Copyright
© EDP Sciences, SMAI, 2011

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