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Optimal convex shapes for concave functionals (2010)
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BibTeX
@MISC{Bucur10optimalconvex,
author = {Dorin Bucur and Ilaria Fragalà and Jimmy Lamboley},
title = {Optimal convex shapes for concave functionals},
year = {2010}
}
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Abstract
Motivated by a long-standing conjecture of Pólya 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 Pólya-Szegö problem.
Keyphrases
optimal convex shape concave functionals convex body large class several counterexample dirichlet energy shape optimization second order shape derivative surface measure concavity inequality convex portion newtonian capacity lya-szeg problem kneser-s s inequality long-standing conjecture variational functionals positive gauss curvature new algebraic structure indecomposability result quantitative version global concavity optimal shape local analysis smooth region
