Optimal convex shapes for concave functionals

Dorin Bucur; Ilaria Fragalà; Jimmy Lamboley

ESAIM: Control, Optimisation and Calculus of Variations (2012)

  • Volume: 18, Issue: 3, page 693-711
  • ISSN: 1292-8119

Abstract

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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 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 Pólya-Szegö problem.

How to cite

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Bucur, Dorin, Fragalà, Ilaria, and Lamboley, Jimmy. "Optimal convex shapes for concave functionals." ESAIM: Control, Optimisation and Calculus of Variations 18.3 (2012): 693-711. <http://eudml.org/doc/277816>.

@article{Bucur2012,
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 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 Pólya-Szegö problem. },
author = {Bucur, Dorin, Fragalà, Ilaria, Lamboley, Jimmy},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
keywords = {Convex bodies; concavity inequalities; optimization; shape derivatives; capacity; convex bodies},
language = {eng},
month = {11},
number = {3},
pages = {693-711},
publisher = {EDP Sciences},
title = {Optimal convex shapes for concave functionals},
url = {http://eudml.org/doc/277816},
volume = {18},
year = {2012},
}

TY - JOUR
AU - Bucur, Dorin
AU - Fragalà, Ilaria
AU - Lamboley, Jimmy
TI - Optimal convex shapes for concave functionals
JO - ESAIM: Control, Optimisation and Calculus of Variations
DA - 2012/11//
PB - EDP Sciences
VL - 18
IS - 3
SP - 693
EP - 711
AB - 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 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 Pólya-Szegö problem.
LA - eng
KW - Convex bodies; concavity inequalities; optimization; shape derivatives; capacity; convex bodies
UR - http://eudml.org/doc/277816
ER -

References

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