Partial Differential Equations/Geometry
Euclidean balls solve some isoperimetric problems with nonradial weights
Comptes Rendus. Mathématique, Volume 350 (2012) no. 21-22, pp. 945-947.

In this Note we present the solution of some isoperimetric problems in open convex cones of Rn 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.

Dans cette Note, nous présentons la solution de certains problèmes isopérimétriques dans des cônes convexes de Rn 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.

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DOI: 10.1016/j.crma.2012.10.031
Cabré, Xavier 1, 2; Ros-Oton, Xavier 2; Serra, Joaquim 2

1 ICREA (Institució Catalana de Recerca i Estudis Avançats), Spain
2 Universitat Politècnica de Catalunya, Departament Matemàtica Aplicada I, Avda. Diagonal 647, 08028 Barcelona, Spain
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Cabré, Xavier; Ros-Oton, Xavier; Serra, Joaquim. 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. http://www.numdam.org/articles/10.1016/j.crma.2012.10.031/

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