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/
[1] , Zur theorie der gemischten volumina von konvexen korpern III, Mat. Sb. 3 (1938) 27-46. | JFM | Zbl
[2] , and , Blaschke addition and convex polyedra. preprint, arXiv:math/0502345 (2005). | MR | Zbl
[3] and , Concavity properties for elliptic free boundary problems. Nonlinear Anal. 71 (2009) 4461-4470. | MR | Zbl
[4] , Capacitary inequalities of the Brunn-Minkowki type. Math. Ann. 263 (1983) 179-184. | MR | Zbl
[5] , Greenian potentials and concavity. Math. Ann. 272 (1985) 155-160. | MR | Zbl
[6] and , On extension of the Brunn-Minkowski and Prékopa-Leindler inequality, including inequalities for log concave functions, and with an application to diffision equation. J. Funct. Anal. 22 (1976) 366-389. | MR | Zbl
[7] , and , A Symmetry Problem in the Calculus of Variations, Calc. Var. Partial Differential Equations 4 (1996) 593-599. | MR | Zbl
[8] , Extremal H-convex bodies. Sibirsk Mat. Zh. 20 (1979) 412-415. | MR | Zbl
[9] , and , Minimization of λ2(Ω) with a perimeter constraint. Indiana Univ. Math. J. 58 (2009) 2709-2728. | MR | Zbl
[10] , and , On the case of equality in the Brunn-Minkowski inequality for capacity. Adv. Math. 117 (1996) 193-207. | MR | Zbl
[11] and , On volume product inequalities for convex sets. Proc. Amer. Math. Soc. 134 (2006) 2393-2402. | MR | Zbl
[12] , Brunn-Minkowski inequalities for variational functionals and related problems. Adv. Math. 194 (2005) 105-140. | MR | Zbl
[13] and , The Brunn-Minkowski inequality for the n-dimensional logarithmic capacity. Potential Anal. 22 (2005) 289-304 | MR | Zbl
[14] and , The Minkowski problem for the torsional rigidity. Indiana Univ. Math. J. 59 (2010) 1013-1040. | MR | Zbl
[15] and , The Brunn-Minkowski inequality for p-capacity of convex bodies. Math. Ann. 327 (2003) 459-479. | MR | Zbl
[16] and , Some estimates of the minimizing properties of web functions, Calc. Var. Partial Differential Equations 15 (2002) 45-66. | MR | Zbl
[17] , and , On a long-standing conjecture by Pólya-Szegö and related topics. Z. Angew. Math. Phys. 56 (2005) 763-782. | MR | Zbl
[18] , and , A refined Brunn-Minkowski inequality for convex sets. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26 (2009) 2511-2519. | Numdam | MR | Zbl
[19] , and , On an isoperimetric inequality for capacity conjectured by Pólya and Szegö. J. Differ. Equ. 250 (2011) 1500-1520. | MR | Zbl
[20] , Upper and lower bounds for the first Dirichlet eigenvalue of a triangle, Proc. Am. Math. Soc. 134 (2006) 2083-2089. | MR | Zbl
[21] , The Brunn-Minkowski inequality. Bull. Am. Math. Soc. (N.S.) 39 (2002) 355-405. | MR | Zbl
[22] and , Capacities, surface area, and radial sums, Adv. Math. 221 (2009) 601-626. | MR | Zbl
[23] and , On the intermediate area functions of convex bodies. Math. Z. 173 (1980) 185-194. | MR | Zbl
[24] and , Convolutions, transforms, and convex bodies. Proc. London Math. Soc. 78 (1999) 77-115. | MR | Zbl
[25] , . Monatsh. Math. 59 (1955) 230-237. | MR | Zbl
[26] and , Variation et Optimisation de Formes : une analyse géométrique, Mathématiques et Applications 48. Springer (2005). | MR | Zbl
[27] , A Minkowski problem for electrostatic capacity. Acta Math. 176 (1996) 1-47. | MR | Zbl
[28] , The direct method in the calculus of variations for convex bodies. Adv. Math. 122 (1996) 262-279. | MR | Zbl
[29] , One functional-analytical idea by Alexandrov in convex geometry. Vladikavkaz. Mat. Zh. 4 (2002) 50-55. | MR | Zbl
[30] , Pareto optimality and isoperimetry. preprint, arXiv:0902.1157v1 (2009).
[31] and , An example of non-convex minimization and an application to Newton's problem of the body of least resistance. Ann. Inst. Henri Poincaré 18 (2001) 179-198. | Numdam | MR | Zbl
[32] and , Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim. 48 (2009/10) 3003-3025. | MR | Zbl
[33] , and , Regularity and singularities of optimal convex shapes in the plane, preprint (2011). | MR | Zbl
[34] , , Higher order differentiability properties of the composition and of the inversion operator. Indag. Math. N S 5 (1994) 457-482. | MR | Zbl
[35] and , Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies 27. Princeton University Press, Princeton, N.J. (1951). | MR | Zbl
[36] , Univalent functions. Vandenhoeck and Ruprecht, Göttingen (1975). | MR | Zbl
[37] , A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue. Adv. Math. 194 (2005) 67-86. | MR | Zbl
[38] , Eine allgemeine Extremaleigenschaft der Kugel. Monatsh. Math. 71 (1967) 231-237. | MR | Zbl
[39] , Convex bodies : the Brunn-Minkowski theory. Cambridge Univ. Press (1993). | MR | Zbl
Cité par Sources :
