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.
Mots-clés : convex bodies, concavity inequalities, optimization, shape derivatives, capacity
@article{COCV_2012__18_3_693_0,
author = {Bucur, Dorin and Fragal\`a, Ilaria and Lamboley, Jimmy},
title = {Optimal convex shapes for concave functionals},
journal = {ESAIM: Control, Optimisation and Calculus of Variations},
pages = {693--711},
publisher = {EDP-Sciences},
volume = {18},
number = {3},
year = {2012},
doi = {10.1051/cocv/2011167},
mrnumber = {3041661},
zbl = {1253.49031},
language = {en},
url = {http://www.numdam.org/articles/10.1051/cocv/2011167/}
}
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 PY - 2012 SP - 693 EP - 711 VL - 18 IS - 3 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/cocv/2011167/ DO - 10.1051/cocv/2011167 LA - en ID - COCV_2012__18_3_693_0 ER -
%0 Journal Article %A Bucur, Dorin %A Fragalà, Ilaria %A Lamboley, Jimmy %T Optimal convex shapes for concave functionals %J ESAIM: Control, Optimisation and Calculus of Variations %D 2012 %P 693-711 %V 18 %N 3 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/cocv/2011167/ %R 10.1051/cocv/2011167 %G en %F COCV_2012__18_3_693_0
Bucur, Dorin; Fragalà, Ilaria; Lamboley, Jimmy. Optimal convex shapes for concave functionals. ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 693-711. doi : 10.1051/cocv/2011167. http://www.numdam.org/articles/10.1051/cocv/2011167/
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