Indecomposable sets of finite perimeter in doubling metric measure spaces

Bonicatto, Paolo; Pasqualetto, Enrico; Rajala, Tapio

doi:10.1007/s00526-020-1725-7

Cited by 9 publications

(20 citation statements)

References 28 publications

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“…Recall that ∂ e (E ∩ F ) ⊂ ∂ e E ∪ ∂ e F , see e.g. [16,Proposition 1.16(ii)]. Therefore, we have that…”

Section: Deformation Lemmamentioning

confidence: 98%

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Isoperimetric sets in spaces with lower bounds on the Ricci curvature

Antonelli,

Pasqualetto,

Pozzetta

2021

Preprint

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In this paper we study regularity and topological properties of volume constrained minimizers of quasi-perimeters in RCD spaces where the reference measure is the Hausdorff measure. A quasi-perimeter is a functional given by the sum of the usual perimeter and of a suitable continuous term. In particular, isoperimetric sets are a particular case of our study.We prove that on an RCD(K, N ) space (X, d, H N ), with K ∈ R, N ≥ 2, and a uniform bound from below on the volume of unit balls, volume constrained minimizers of quasi-perimeters are open bounded sets with (N − 1)-Ahlfors regular topological boundary coinciding with the essential boundary.The proof is based on a new Deformation Lemma for sets of finite perimeter in RCD(K, N ) spaces (X, d, m) and on the study of interior and exterior points of volume constrained minimizers of quasi-perimeters.The theory applies to volume constrained minimizers in smooth Riemannian manifolds, possibly with boundary, providing a general regularity result for such minimizers in the smooth setting.

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“…Recall that ∂ e (E ∩ F ) ⊂ ∂ e E ∪ ∂ e F , see e.g. [16,Proposition 1.16(ii)]. Therefore, we have that…”

Section: Deformation Lemmamentioning

confidence: 98%

“…We remark that all RCD(K, N) spaces with N < ∞, see Section 2.2.3 for the definition, are isotropic PI spaces; cf. [16,Example 1.31(iii)].…”

Section: Deformation Lemmamentioning

confidence: 99%

Isoperimetric sets in spaces with lower bounds on the Ricci curvature

Antonelli,

Pasqualetto,

Pozzetta

2021

Preprint

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“…The theory was generalized to metric measure spaces by Bonicatto-Pasqualetto-Rajala [6]. As is common in analysis on metric measure spaces, they assumed the space (X, d, m) to be complete, equipped with a doubling measure, and support a (1, 1)-Poincaré inequality.…”

Section: Introductionmentioning

confidence: 99%

“…in [4, Section 7] and it is satisfied in Euclidean as well as various other PI spaces. However, it excludes some PI spaces from the theory, see [6,Example 1.27].…”

Section: Introductionmentioning

confidence: 99%

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A note on indecomposable sets of finite perimeter

Lahti¹

2021

Preprint

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On the integration of $$L^0$$-Banach $$L^0$$-modules and its applications to vector calculus on $$\textsf{RCD}$$ spaces

Caputo,

Lučić,

Pasqualetto

et al. 2024

Rev Mat Complut

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A finite-dimensional $$\textsf{RCD}$$ RCD space can be foliated into sufficiently regular leaves, where a differential calculus can be performed. Two important examples are given by the measure-theoretic boundary of the superlevel set of a function of bounded variation and the needle decomposition associated to a Lipschitz function. The aim of this paper is to connect the vector calculus on the lower dimensional leaves with the one on the base space. In order to achieve this goal, we develop a general theory of integration of $$L^0$$ L 0 -Banach $$L^0$$ L 0 -modules of independent interest. Roughly speaking, we study how to ‘patch together’ vector fields defined on the leaves that are measurable with respect to the foliation parameter.

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Indecomposable sets of finite perimeter in doubling metric measure spaces

Cited by 9 publications

References 28 publications

Isoperimetric sets in spaces with lower bounds on the Ricci curvature

Isoperimetric sets in spaces with lower bounds on the Ricci curvature

A note on indecomposable sets of finite perimeter

On the integration of $$L^0$$-Banach $$L^0$$-modules and its applications to vector calculus on $$\textsf{RCD}$$ spaces

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