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15
On the isoperimetric problem for radial logconvex densities
 Calc. Var. Partial Differential Equations
"... Abstract. Given a smooth, radial, uniformly logconvex density eV on Rn, n ≥ 2, we characterize isoperimetric sets E with respect to weighted perimeter R ∂E eV dHn−1 and weighted volume m = R E eV as balls centered at the origin, provided m ∈ [0, m0) for some (potentially computable) m0> 0; this ..."
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Cited by 12 (3 self)
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Abstract. Given a smooth, radial, uniformly logconvex density eV on Rn, n ≥ 2, we characterize isoperimetric sets E with respect to weighted perimeter R ∂E eV dHn−1 and weighted volume m = R E eV as balls centered at the origin, provided m ∈ [0, m0) for some (potentially computable) m0> 0; this affirmatively answers conjecture [RCBM, Conjecture 3.12] for such values of the weighted volume parameter. We also prove that the set of weighted volumes such that this characterization holds true is open, thus reducing the proof of the full conjecture to excluding the possibility of bifurcation values of the weighted volume parameter. Finally, we show the validity of the conjecture when V belongs to a C2neighborhood of cx2 (c> 0).
Sharp isoperimetric inequalities via the ABP method
"... We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of Rn. Our result ..."
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Cited by 7 (2 self)
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We prove some old and new isoperimetric inequalities with the best constant using the ABP method applied to an appropriate linear Neumann problem. More precisely, we obtain a new family of sharp isoperimetric inequalities with weights (also called densities) in open convex cones of Rn. Our result applies to all nonnegative homogeneous weights satisfying a concavity condition in the cone. Remarkably, Euclidean balls centered at the origin (intersected with the cone) minimize the weighted isoperimetric quotient, even if all our weights are nonradial —except for the constant ones. We also study the anisotropic isoperimetric problem in convex cones for the same class of weights. We prove that the Wulff shape (intersected with the cone) minimizes the anisotropic weighted perimeter under the weighted volume constraint. As a particular case of our results, we give new proofs of two classical results: the Wulff inequality and the isoperimetric inequality in convex cones of Lions and Pacella.
Euclidean balls solve some isoperimetric problems with nonradial weights
"... Abstract. 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 isoperim ..."
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Cited by 5 (2 self)
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Abstract. 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. Résumé. Dans cette note, nous présentons la solution de certains problèmes isopérimétriques dans des cônes convexes de Rn ou ̀ le périmètre et le volume sont mesurés par rapport a ̀ certains poids non radiaux. Contrairement a ̀ ce que l’on pourrait penser, les boules euclidiennes centrées a ̀ l’origine (intersectées avec le cône) minimisent le quotient isopérimétrique. Notre résultat s’applique aux poids strictement positifs, homogènes et satisfaisant une condition de concavite ́ dans le cône. Lorsque le poids est constant, le résultat a éte ́ établi par Lions et Pacella en 1990. 1.
Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities
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Complemented Brunn–Minkowski Inequalities and Isoperimetry for Homogeneous and NonHomogeneous Measures
, 2014
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The ε− εβ property, the boundedness of isoperimetric sets in Rn with density, and some applications
 J. Reine Angew. Math. (Crelle
"... Abstract. We show that every isoperimetric set in RN with density is bounded if the density is continuous and bounded by above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we ..."
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Cited by 2 (1 self)
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Abstract. We show that every isoperimetric set in RN with density is bounded if the density is continuous and bounded by above and below. This improves the previously known boundedness results, which basically needed a Lipschitz assumption; on the other hand, the present assumption is sharp, as we show with an explicit example. To obtain our result, we observe that the main tool which is often used, namely a classical “ε − ε ” property already discussed by Allard, Almgren and Bombieri, admits a weaker counterpart which is still sufficient for the boundedness, namely, an “ε − εβ ” version of the property. And in turn, while for the validity of the first property the Lipschitz assumption is essential, for the latter the sole continuity is enough. We conclude by deriving some consequences of our result about the existence and regularity of isoperimetric sets. 1.
Free boundary stable hypersurfaces in manifolds with density and rigidity
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"... This manual is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This manual is distributed in the hope that it will be useful ..."
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This manual is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This manual is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details. You should have received a copy of the GNU General Public License along with this manual if not, write to the Free Software