Isoperimetric Regions in $\mathbb{R}^n$ with density $r^p$
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In this article, it was shown that the unique isoperimetric hypersurfaces with density ρ r^p for ρ > 0 are spheres that pass through the origin.Abstract:
We show that the unique isoperimetric hypersurfaces in $\mathbb{R}^n$ with density $r^p$ for $n \ge 3$ and $p>0$ are spheres that pass through the origin.read more
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The ε - εβ Property in the Isoperimetric Problem with Double Density, and the Regularity of Isoperimetric Sets
TL;DR: In this article, the authors proved the validity of the ε - ε β {varepsilon-β {β − VAREPSIL-VAREPSI-β} property in the isoperimetric problem with double density.
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The $\varepsilon-\varepsilon^\beta$ property in the isoperimetric problem with double density, and the regularity of isoperimetric sets
TL;DR: In this article, the authors proved the validity of the regularity property of isoperimetric sets with double density, generalising the known properties for the case of single density.
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Some isoperimetric inequalities on $\mathbb{R} ^N$ with respect to weights $|x|^\alpha $
TL;DR: In this paper, a class of isoperimetric problems with respect to weights that are powers of the distance to the origin were solved, and the radiality of optimizers in some Caffarelli-Kohn-Nirenberg inequalities was established.
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On the isoperimetric problem with perimeter density r^p
TL;DR: In this paper, the authors studied the isoperimetric problem in the case of perimeter density and volume density in the range $n = 2,$ and showed that the results in this range do not generalize for the range of $n+1+1
References
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TL;DR: Geometric measure theory can be described as differential geometry, generalized through measure theory to deal with maps and surfaces that are not necessarily smooth and applied to the calculus of variations as mentioned in this paper.
Journal ArticleDOI
On the isoperimetric problem in Euclidean space with density
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TL;DR: In this article, the authors add to the literature the well-known fact that an isoperimetric hypersurface S of dimension at most six in a smooth Riemannian manifold M is a smooth submanifold.
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Isoperimetric problems in sectors with density
TL;DR: In this paper, the authors considered the problem of isoperimetric problem in planar sectors with density ρ, ρ = ρ −1/ρ, and with density α > 1/α inside the unit disk and ρ ≥ 1$ outside.
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Isoperimetric regions in log-convex densities
TL;DR: In this paper, the authors proved that balls around the origin constitute isoperimetric regions of any given volume, proving the Log-Convex Density Conjecture due to Kenneth Brakke.