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SOBOLEV AND ISOPERIMETRIC INEQUALITIES WITH MONOMIAL WEIGHTS
"... Abstract. We consider the monomial weight x1A1 · · · xnAn in Rn, where Ai ≥ 0 is a real number for each i = 1,..., n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replace ..."
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Cited by 9 (4 self)
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Abstract. We consider the monomial weight x1A1 · · · xnAn in Rn, where Ai ≥ 0 is a real number for each i = 1,..., n, and establish Sobolev, isoperimetric, Morrey, and Trudinger inequalities involving this weight. They are the analogue of the classical ones with the Lebesgue measure dx replaced by x1A1 · · · xnAndx, and they contain the best or critical exponent (which depends on A1,..., An). More importantly, for the Sobolev and isoperimetric inequalities, we obtain the best constant and extremal functions. When Ai are nonnegative integers, these inequalities are exactly the classical ones in the Euclidean space RD (with no weight) when written for axially symmetric functions and domains in RD = RA1+1 × · · · × RAn+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.
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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EXISTENCE OF ISOPERIMETRIC SETS WITH DENSITIES “CONVERGING FROM BELOW ” ON RN
"... Abstract. In this paper, we consider the isoperimetric problem in the space RN with density. Our result states that, if the density f is l.s.c. and converges to a limit a> 0 at infinity, being f ≤ a far from the origin, then isoperimetric sets exist for all volumes. Several known results or count ..."
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Abstract. In this paper, we consider the isoperimetric problem in the space RN with density. Our result states that, if the density f is l.s.c. and converges to a limit a> 0 at infinity, being f ≤ a far from the origin, then isoperimetric sets exist for all volumes. Several known results or counterexamples show that the present result is essentially sharp. The special case of our result for radial and increasing densities posively answers a conjecture made in [10]. 1.