Optimal convex shapes for concave functionals

Bucur, Dorin  ; Fragalà, Ilaria  ; Lamboley, Jimmy 

ESAIM: Control, Optimisation and Calculus of Variations, Tome 18 (2012) no. 3, pp. 693-711.

Résumé

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.

Détail

DOI : 10.1051/cocv/2011167

Classification : 49Q10, 31A15
Keywords: convex bodies, concavity inequalities, optimization, shape derivatives, capacity

Bibliographie

[1] A.D. Alexandrov, Zur theorie der gemischten volumina von konvexen korpern III, Mat. Sb. 3 (1938) 27-46. | Zbl | JFM

[2] V. Alexandrov, N. Kopteva and S.S. Kutateladze, Blaschke addition and convex polyedra. preprint, arXiv:math/0502345 (2005). | Zbl | MR

[3] C. Bianchini and P. Salani, Concavity properties for elliptic free boundary problems. Nonlinear Anal. 71 (2009) 4461-4470. | Zbl | MR

[4] C. Borell, Capacitary inequalities of the Brunn-Minkowki type. Math. Ann. 263 (1983) 179-184. | Zbl | MR

[5] C. Borell, Greenian potentials and concavity. Math. Ann. 272 (1985) 155-160. | Zbl | MR

[6] H. Brascamp and E. Lieb, 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. | Zbl | MR

[7] F. Brock, V. Ferone and B. Kawohl, A Symmetry Problem in the Calculus of Variations, Calc. Var. Partial Differential Equations 4 (1996) 593-599. | Zbl | MR

[8] E.M. Bronshtein, Extremal H-convex bodies. Sibirsk Mat. Zh. 20 (1979) 412-415. | Zbl | MR

[9] D. Bucur, G. Buttazzo and A. Henrot, Minimization of λ2(Ω) with a perimeter constraint. Indiana Univ. Math. J. 58 (2009) 2709-2728. | Zbl | MR

[10] L. Caffarelli, D. Jerison and E. Lieb, On the case of equality in the Brunn-Minkowski inequality for capacity. Adv. Math. 117 (1996) 193-207. | Zbl | MR

[11] S. Campi and P. Gronchi, On volume product inequalities for convex sets. Proc. Amer. Math. Soc. 134 (2006) 2393-2402. | Zbl | MR

[12] A. Colesanti, Brunn-Minkowski inequalities for variational functionals and related problems. Adv. Math. 194 (2005) 105-140. | Zbl | MR

[13] A. Colesanti and P. Cuoghi, The Brunn-Minkowski inequality for the n-dimensional logarithmic capacity. Potential Anal. 22 (2005) 289-304 | Zbl | MR

[14] A. Colesanti and M. Fimiani, The Minkowski problem for the torsional rigidity. Indiana Univ. Math. J. 59 (2010) 1013-1040. | Zbl | MR

[15] A. Colesanti and P. Salani, The Brunn-Minkowski inequality for p-capacity of convex bodies. Math. Ann. 327 (2003) 459-479. | Zbl | MR

[16] G. Crasta and F. Gazzola, Some estimates of the minimizing properties of web functions, Calc. Var. Partial Differential Equations 15 (2002) 45-66. | Zbl | MR

[17] G. Crasta, I. Fragalà and F. Gazzola, On a long-standing conjecture by Pólya-Szegö and related topics. Z. Angew. Math. Phys. 56 (2005) 763-782. | Zbl | MR

[18] A. Figalli, F. Maggi and A. Pratelli, A refined Brunn-Minkowski inequality for convex sets. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26 (2009) 2511-2519. | Zbl | MR | mathdoc-id

[19] I. Fragalà, F. Gazzola and M. Pierre, On an isoperimetric inequality for capacity conjectured by Pólya and Szegö. J. Differ. Equ. 250 (2011) 1500-1520. | Zbl | MR

[20] P. Freitas, Upper and lower bounds for the first Dirichlet eigenvalue of a triangle, Proc. Am. Math. Soc. 134 (2006) 2083-2089. | Zbl | MR

[21] R. Gardner, The Brunn-Minkowski inequality. Bull. Am. Math. Soc. (N.S.) 39 (2002) 355-405. | Zbl | MR

[22] R.J. Gardner and D. Hartenstine, Capacities, surface area, and radial sums, Adv. Math. 221 (2009) 601-626. | Zbl | MR

[23] P.R. Goodey and R. Schneider, On the intermediate area functions of convex bodies. Math. Z. 173 (1980) 185-194. | Zbl | MR

[24] E. Grinberg and G. Zhang, Convolutions, transforms, and convex bodies. Proc. London Math. Soc. 78 (1999) 77-115. | Zbl | MR

[25] H. Hadwiger, Konkave Eikörperfunktionale. Monatsh. Math. 59 (1955) 230-237. | Zbl | MR

[26] A. Henrot and M. Pierre, Variation et Optimisation de Formes : une analyse géométrique, Mathématiques et Applications 48. Springer (2005). | Zbl | MR

[27] D. Jerison, A Minkowski problem for electrostatic capacity. Acta Math. 176 (1996) 1-47. | Zbl | MR

[28] D. Jerison, The direct method in the calculus of variations for convex bodies. Adv. Math. 122 (1996) 262-279. | Zbl | MR

[29] S.S. Kutateladze, One functional-analytical idea by Alexandrov in convex geometry. Vladikavkaz. Mat. Zh. 4 (2002) 50-55. | Zbl | MR

[30] S.S. Kutateladze, Pareto optimality and isoperimetry. preprint, arXiv:0902.1157v1 (2009).

[31] T. Lachand-Robert and M.A. Peletier, 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. | Zbl | MR | mathdoc-id

[32] J. Lamboley and A. Novruzi, Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim. 48 (2009/10) 3003-3025. | Zbl | MR

[33] J. Lamboley, A. Novruzi and M. Pierre, Regularity and singularities of optimal convex shapes in the plane, preprint (2011). | Zbl | MR

[34] M. Lanza, De Cristoforis, Higher order differentiability properties of the composition and of the inversion operator. Indag. Math. N S 5 (1994) 457-482. | Zbl | MR

[35] G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies 27. Princeton University Press, Princeton, N.J. (1951). | Zbl | MR

[36] Ch. Pommerenke, Univalent functions. Vandenhoeck and Ruprecht, Göttingen (1975). | Zbl | MR

[37] P. Salani, A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue. Adv. Math. 194 (2005) 67-86. | Zbl | MR

[38] R. Schneider, Eine allgemeine Extremaleigenschaft der Kugel. Monatsh. Math. 71 (1967) 231-237. | Zbl | MR

[39] R. Schneider, Convex bodies : the Brunn-Minkowski theory. Cambridge Univ. Press (1993). | Zbl | MR