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12
ISOPERIMETRY AND STABILITY PROPERTIES OF BALLS WITH RESPECT TO NONLOCAL ENERGIES
, 2014
"... Abstract. We obtain a sharp quantitative isoperimetric inequality for nonlocal sperimeters, uniform with respect to s bounded away from 0. This allows us to address local and global minimality properties of balls with respect to the volumeconstrained minimization of a free energy consisting of a n ..."
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Cited by 12 (4 self)
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Abstract. We obtain a sharp quantitative isoperimetric inequality for nonlocal sperimeters, uniform with respect to s bounded away from 0. This allows us to address local and global minimality properties of balls with respect to the volumeconstrained minimization of a free energy consisting of a nonlocal sperimeter plus a nonlocal repulsive interaction term. In the particular case s = 1 the sperimeter coincides with the classical perimeter, and our results improve the ones of Knüpfer and Muratov [25, 26] concerning minimality of balls of small volume in isoperimetric problems with a competition between perimeter and a nonlocal potential term. More precisely, their result is extended to its maximal range of validity concerning the type of nonlocal potentials considered, and is also generalized to the case where local perimeters are replaced by their nonlocal counterparts. 1.
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.
Trombetti: On a conjectured reverse FaberKrahn inequality for a Steklovtype Laplacian eigenvalue
, 2013
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Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities
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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.
STABILITY RESULTS FOR THE BRUNNMINKOWSKI INEQUALITY
"... The BrunnMiknowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This classical inequality in convex geometry was inspired by issues around the isoperimetric problem and was considered for a long time to belong to geometry, where ..."
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The BrunnMiknowski inequality gives a lower bound on the Lebesgue measure of a sumset in terms of the measures of the individual sets. This classical inequality in convex geometry was inspired by issues around the isoperimetric problem and was considered for a long time to belong to geometry, where its significance is widely recognized. However, it is by now clear that
PROOF OF THE LOGCONVEX DENSITY CONJECTURE
"... Abstract. We completely characterize isoperimetric regions in Rn with density eh, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the LogConvex Density Conjecture due to Kenneth Brakke. ..."
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Abstract. We completely characterize isoperimetric regions in Rn with density eh, where h is convex, smooth, and radially symmetric. In particular, balls around the origin constitute isoperimetric regions of any given volume, proving the LogConvex Density Conjecture due to Kenneth Brakke.