Calculus of Variations and Geometric Measure Theory

D. Bucur - I. Fragalà - J. Lamboley

Optimal convex shapes for concave functionals

created by lamboley on 26 May 2016


Published Paper

Inserted: 26 may 2016
Last Updated: 26 may 2016

Journal: ESAIM Control Optim. Calc. Var.
Volume: 18
Number: 3
Pages: 693-711
Year: 2012


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 isoperimetric-like 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.