Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities

Cañete, Antonio; Rosales, César

doi:10.48550/arxiv.1304.1438

Cited by 3 publications

(11 citation statements)

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“…On weighted manifolds, Cañete and Rosales [3] showed the rigidity of compact stable hypersurfaces with free boundary in a convex solid cone in Euclidean space with homogeneous density. Our next result extends Choe and Park's result to weighted Euclidean spaces (R n+1 , ds 0 , dμ), where ds 0 is the Euclidean metric.…”

Section: Introductionmentioning

confidence: 99%

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

Batista

Cavalcante

Pyo

2014

Journal of Mathematical Analysis and Applications

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Section: Introductionmentioning

confidence: 99%

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

Batista

Cavalcante

Pyo

2014

Journal of Mathematical Analysis and Applications

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“…1.5] that such spherical caps are f -stable (they are minimizers of the interior weighted area for fixed weighted volume). In fact, in [9,Thm. 5.11] it is shown that these are the unique compact f -stable hypersurfaces in M .…”

Section: Stationary and Stable Free Boundary Hypersurfacesmentioning

confidence: 99%

“…Let M be a Euclidean solid cone endowed with a k-homogeneous density f = e ψ . In [9,Ex. 4.7] it was shown that the intersection with M of a round sphere centered at 0 is strongly f -stable if and only if k −n.…”

Section: Stationary and Stable Free Boundary Hypersurfacesmentioning

confidence: 99%

“…Consider the manifold M r = {p ∈ R 3 ; |p| r} endowed with the Euclidean metric and the radial k-homogeneous density f (p) = |p| k . It is easy to check, see the computations in [9,Ex. 4.3], that the f -mean curvature of ∂M r with respect to the inner unit normal equals −(k + 2)/r.…”

Section: Non-negative Bakry-émery-ricci Curvature and Locally Convex ...mentioning

confidence: 99%

“…However, besides showing some relevant situations in Examples 3.4, 3.7 and 3.8, the partitioning problem and the associated f -stable hypersurfaces will not be treated in detail. Some characterization results for compact f -stable hypersurfaces with free boundary in a Euclidean solid cone where a homogeneous density is considered can be found in [9].…”

Section: Introductionmentioning

confidence: 99%

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Free boundary stable hypersurfaces in manifolds with density and rigidity results

Castro

Rosales

2014

Journal of Geometry and Physics

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Abstract. Let M be a weighted manifold with boundary ∂M , i.e., a Riemannian manifold where a density function is used to weight the Riemannian Hausdorff measures. In this paper we compute the first and the second variational formulas of the interior weighted area for deformations by hypersurfaces with boundary in ∂M . As a consequence, we obtain variational characterizations of critical points and second order minima of the weighted area with or without a volume constraint. Moreover, in the compact case, we obtain topological estimates and rigidity properties for free boundary stable and area-minimizing hypersurfaces under certain curvature and boundary assumptions on M . Our results and proofs extend previous ones for Riemannian manifolds (constant densities) and for hypersurfaces with empty boundary in weighted manifolds.

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Sharp isoperimetric inequalities via the ABP method

Cabré¹,

Ros‐Oton²,

Serra³

2016

J. Eur. Math. Soc.

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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 R n . 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.2010 Mathematics Subject Classification. Primary 28A75; Secondary 35A23, 49Q20.

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Compact stable hypersurfaces with free boundary in convex solid cones with homogeneous densities

Cited by 3 publications

References 0 publications

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

Some isoperimetric inequalities and eigenvalue estimates in weighted manifolds

Free boundary stable hypersurfaces in manifolds with density and rigidity results

Sharp isoperimetric inequalities via the ABP method

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