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Isoperimetric regions in the plane with density r^p
, 2010
"... We consider the isoperimetric problem in the plane with density r p, p> 0, and prove that the solution is a circle through the origin. We use the stability of this isoperimetric curve to prove an ..."
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Cited by 12 (0 self)
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We consider the isoperimetric problem in the plane with density r p, p> 0, and prove that the solution is a circle through the origin. We use the stability of this isoperimetric curve to prove an
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).
Manifolds with density, applications and gradient Schrödinger operators
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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.
On the isoperimetric problem with respect to a mixed EuclideanGaussian density
 J. Funct. Anal
"... Abstract. The isoperimetric problem with respect to the producttype density e − x 2 2 dx dy on the Euclidean space R h × R k is studied, with particular emphasis on the case k = 1. A conjecture about the minimality of large cylinders in the case k > 1 is also formulated. ..."
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Cited by 6 (2 self)
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Abstract. The isoperimetric problem with respect to the producttype density e − x 2 2 dx dy on the Euclidean space R h × R k is studied, with particular emphasis on the case k = 1. A conjecture about the minimality of large cylinders in the case k > 1 is also formulated.
On isoperimetric sets of radially symmetric measures
 CONCENTRATION, FUNCTIONAL INEQUALITIES AND ISOPERIMETRY (PROC. INTL. WKSHP., FLORIDA ATLANTIC UNIV., OCT./NOV. 2009), NUMBER 545 IN CONTEMPORARY MATHEMATICS
, 2011
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