[Les boules euclidiennes minimisent certains problèmes isopérimétriques avec des poids non radiaux]
Dans cette Note, nous présentons la solution de certains problèmes isopérimétriques dans des cônes convexes de où le périmètre et le volume sont mesurés par rapport à certains poids non radiaux. Contrairement à ce que lʼon pourrait penser, les boules euclidiennes centrées à lʼorigine (intersectées avec le cône) minimisent le quotient isopérimétrique. Notre résultat sʼapplique aux poids strictement positifs, homogènes et satisfaisant une condition de concavité dans le cône. Lorsque le poids est constant, le résultat a été établi par Lions et Pacella en 1990.
In this Note we present the solution of some isoperimetric problems in open convex cones of in which perimeter and volume are measured with respect to certain nonradial weights. Surprisingly, Euclidean balls centered at the origin (intersected with the convex cone) minimize the isoperimetric quotient. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. When the weight is constant, the result was established by Lions and Pacella in 1990.
Accepté le :
Publié le :
Xavier Cabré 1, 2 ; Xavier Ros-Oton 2 ; Joaquim Serra 2
@article{CRMATH_2012__350_21-22_945_0,
author = {Xavier Cabr\'e and Xavier Ros-Oton and Joaquim Serra},
title = {Euclidean balls solve some isoperimetric problems with nonradial weights},
journal = {Comptes Rendus. Math\'ematique},
pages = {945--947},
publisher = {Elsevier},
volume = {350},
number = {21-22},
year = {2012},
doi = {10.1016/j.crma.2012.10.031},
language = {en},
}
TY - JOUR AU - Xavier Cabré AU - Xavier Ros-Oton AU - Joaquim Serra TI - Euclidean balls solve some isoperimetric problems with nonradial weights JO - Comptes Rendus. Mathématique PY - 2012 SP - 945 EP - 947 VL - 350 IS - 21-22 PB - Elsevier DO - 10.1016/j.crma.2012.10.031 LA - en ID - CRMATH_2012__350_21-22_945_0 ER -
Xavier Cabré; Xavier Ros-Oton; Joaquim Serra. Euclidean balls solve some isoperimetric problems with nonradial weights. Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 945-947. doi : 10.1016/j.crma.2012.10.031. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2012.10.031/
[1] Weighted isoperimetric inequalities in cones and applications, Nonlinear Anal., Volume 75 (2012), pp. 5737-5755
[2] Partial differential equations, geometry, and stochastic control, Butl. Soc. Catalana Mat., Volume 15 (2000), pp. 7-27 (in Catalan)
[3] Elliptic PDEs in Probability and Geometry. Symmetry and regularity of solutions, Discrete Contin. Dyn. Syst., Volume 20 (2008), pp. 425-457
[4] X. Cabré, X. Ros-Oton, Regularity of stable solutions up to dimension 7 in domains of double revolution, Comm. Partial Differential Equations, in press, . | DOI
[5] Sobolev and isoperimetric inequalities with monomial weights | arXiv
[6] X. Cabré, X. Ros-Oton, J. Serra, Sharp isoperimetric inequalities via the ABP method, in preparation.
[7] A. Díaz, N. Harman, S. Howe, D. Thompson, Isoperimetric problem in sectors with density, Adv. Geom., in press, . | DOI
[8] Isoperimetric inequality for convex cones, Proc. Amer. Math. Soc., Volume 109 (1990), pp. 477-485
[9] Sharp estimates for solutions to a certain type of singular elliptic boundary value problems in two dimensions, Applicable Anal., Volume 12 (1981), pp. 307-321
[10] Manifolds with density, Notices Amer. Math. Soc., Volume 52 (2005), pp. 853-858
Cité par Sources :
Commentaires - Politique