Hostname: page-component-586b7cd67f-r5fsc Total loading time: 0 Render date: 2024-11-28T01:09:08.809Z Has data issue: false hasContentIssue false
  • English
  • Français

Optimal convex shapes for concave functionals

Published online by Cambridge University Press:  29 September 2011

Dorin Bucur ,
Ilaria Fragalà  and
Jimmy Lamboley
Dorin Bucur
Affiliation:
Laboratoire de Mathématiques UMR 5127, Université de Savoie, Campus Scientifique, 73376 Le-Bourget-du-Lac, France
Ilaria Fragalà
Affiliation:
Dipartimento di Matematica, Politecnico, Piazza Leonardo da Vinci, 32, 20133 Milano, Italy. [email protected]
Jimmy Lamboley
Affiliation:
Ceremade UMR 7534, Université de Paris-Dauphine, Place du Maréchal De Lattre De Tassigny, 75775 Paris Cedex 16, France
Get access

Abstract

Motivated by a long-standing conjecture of Pólya and Szegö about the Newtonian capacityof convex bodies, we discuss the role of concavity inequalities in shape optimization, andwe provide several counterexamples to the Blaschke-concavity of variational functionals,including capacity. We then introduce a new algebraic structure on convex bodies, whichallows to obtain global concavity and indecomposability results, and we discuss theirapplication to isoperimetric-like inequalities. As a byproduct of this approach we alsoobtain a quantitative version of the Kneser-Süss inequality. Finally, for a large class offunctionals involving Dirichlet energies and the surface measure, we perform a localanalysis of strictly convex portions of the boundary via second ordershape derivatives. This allows in particular to exclude the presence of smooth regionswith positive Gauss curvature in an optimal shape for Pólya-Szegö problem.

Keywords

Convex bodiesconcavity inequalitiesoptimizationshape derivativescapacity
Type
Research Article
Information
ESAIM: Control, Optimisation and Calculus of Variations , Volume 18 , Issue 3 , July 2012 , pp. 693 - 711
Copyright
© EDP Sciences, SMAI, 2011

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Alexandrov, A.D., Zur theorie der gemischten volumina von konvexen korpern III, Mat. Sb. 3 (1938) 27–46. Google Scholar
V. Alexandrov, N. Kopteva and S.S. Kutateladze, Blaschke addition and convex polyedra. preprint, arXiv:math/0502345 (2005).
Bianchini, C. and Salani, P., Concavity properties for elliptic free boundary problems. Nonlinear Anal. 71 (2009) 4461–4470. Google Scholar
Borell, C., Capacitary inequalities of the Brunn-Minkowki type. Math. Ann. 263 (1983) 179–184. Google Scholar
Borell, C., Greenian potentials and concavity. Math. Ann. 272 (1985) 155–160. Google Scholar
Brascamp, H. and Lieb, E., 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. Google Scholar
Brock, F., Ferone, V. and Kawohl, B., A Symmetry Problem in the Calculus of Variations, Calc. Var. Partial Differential Equations 4 (1996) 593–599. Google Scholar
Bronshtein, E.M., Extremal H-convex bodies. Sibirsk Mat. Zh. 20 (1979) 412–415. Google Scholar
Bucur, D., Buttazzo, G. and Henrot, A., Minimization of λ 2(Ω) with a perimeter constraint. Indiana Univ. Math. J. 58 (2009) 2709–2728. Google Scholar
Caffarelli, L., Jerison, D. and Lieb, E., On the case of equality in the Brunn-Minkowski inequality for capacity. Adv. Math. 117 (1996) 193–207. Google Scholar
Campi, S. and Gronchi, P., On volume product inequalities for convex sets. Proc. Amer. Math. Soc. 134 (2006) 2393–2402. Google Scholar
Colesanti, A., Brunn-Minkowski inequalities for variational functionals and related problems. Adv. Math. 194 (2005) 105–140. Google Scholar
Colesanti, A. and Cuoghi, P., The Brunn-Minkowski inequality for the n-dimensional logarithmic capacity. Potential Anal. 22 (2005) 289–304 Google Scholar
Colesanti, A. and Fimiani, M., The Minkowski problem for the torsional rigidity. Indiana Univ. Math. J. 59 (2010) 1013–1040. Google Scholar
Colesanti, A. and Salani, P., The Brunn-Minkowski inequality for p-capacity of convex bodies. Math. Ann. 327 (2003) 459–479. Google Scholar
Crasta, G. and Gazzola, F., Some estimates of the minimizing properties of web functions, Calc. Var. Partial Differential Equations 15 (2002) 45–66. Google Scholar
Crasta, G., Fragalà, I. and Gazzola, F., On a long-standing conjecture by Pólya-Szegö and related topics. Z. Angew. Math. Phys. 56 (2005) 763–782. Google Scholar
Figalli, A., Maggi, F. and Pratelli, A., A refined Brunn-Minkowski inequality for convex sets. Ann. Inst. Henri Poincaré Anal. Non Linéaire 26 (2009) 2511–2519. Google Scholar
Fragalà, I., Gazzola, F. and Pierre, M., On an isoperimetric inequality for capacity conjectured by Pólya and Szegö. J. Differ. Equ. 250 (2011) 1500–1520. Google Scholar
Freitas, P., Upper and lower bounds for the first Dirichlet eigenvalue of a triangle, Proc. Am. Math. Soc. 134 (2006) 2083–2089. Google Scholar
Gardner, R., The Brunn-Minkowski inequality. Bull. Am. Math. Soc. (N.S.) 39 (2002) 355–405. Google Scholar
Gardner, R.J. and Hartenstine, D., Capacities, surface area, and radial sums, Adv. Math. 221 (2009) 601–626. Google Scholar
Goodey, P.R. and Schneider, R., On the intermediate area functions of convex bodies. Math. Z. 173 (1980) 185–194. Google Scholar
Grinberg, E. and Zhang, G., Convolutions, transforms, and convex bodies. Proc. London Math. Soc. 78 (1999) 77–115. Google Scholar
Hadwiger, H., Konkave Eikörperfunktionale. Monatsh. Math. 59 (1955) 230–237. Google Scholar
A. Henrot and M. Pierre, Variation et Optimisation de Formes : une analyse géométrique, Mathématiques et Applications 48. Springer (2005).
Jerison, D., A Minkowski problem for electrostatic capacity. Acta Math. 176 (1996) 1–47. Google Scholar
Jerison, D., The direct method in the calculus of variations for convex bodies. Adv. Math. 122 (1996) 262–279. Google Scholar
Kutateladze, S.S., One functional-analytical idea by Alexandrov in convex geometry. Vladikavkaz. Mat. Zh. 4 (2002) 50–55. Google Scholar
S.S. Kutateladze, Pareto optimality and isoperimetry. preprint, arXiv:0902.1157v1 (2009).
Lachand-Robert, T. and Peletier, M.A., 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. Google Scholar
Lamboley, J. and Novruzi, A., Polygons as optimal shapes with convexity constraint. SIAM J. Control Optim. 48 (2009/10) 3003–3025. Google Scholar
J. Lamboley, A. Novruzi and M. Pierre, Regularity and singularities of optimal convex shapes in the plane, preprint (2011).
Lanza, M., de Cristoforis, Higher order differentiability properties of the composition and of the inversion operator. Indag. Math. N S 5 (1994) 457–482. Google Scholar
G. Pólya and G. Szegö, Isoperimetric Inequalities in Mathematical Physics, Annals of Mathematics Studies 27. Princeton University Press, Princeton, N.J. (1951).
Ch. Pommerenke, Univalent functions. Vandenhoeck and Ruprecht, Göttingen (1975).
Salani, P., A Brunn-Minkowski inequality for the Monge-Ampère eigenvalue. Adv. Math. 194 (2005) 67–86. Google Scholar
Schneider, R., Eine allgemeine Extremaleigenschaft der Kugel. Monatsh. Math. 71 (1967) 231–237. Google Scholar
R. Schneider, Convex bodies : the Brunn-Minkowski theory. Cambridge Univ. Press (1993).