On the Existence and Nonexistence of Isoperimetric Inequalities with Different Monomial Weights

Abreu, Emerson; Fernandes, Leandro G.

doi:10.1007/s00041-021-09900-8

Cited by 2 publications

(3 citation statements)

References 27 publications

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“…We have already shown that properties (P.1)-(P.5) and (P.7) hold true. The isoperimetric property is discussed (for Lipschitz sets) for monomial weights in [2,7,38,39,64], and for radial weights in [6,68,69]. The Cheeger problem in the monomial setting has also been considered in [7,26].…”

Section: Monomial and Radial Weightsmentioning

confidence: 99%

The Cheeger problem in abstract measure spaces

Franceschi,

Pinamonti,

Saracco

et al. 2023

Journal of London Math Soc

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We consider nonnegative ‐finite measure spaces coupled with a proper functional that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.

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Section: Monomial and Radial Weightsmentioning

confidence: 99%

The Cheeger problem in abstract measure spaces

Franceschi,

Pinamonti,

Saracco

et al. 2023

Journal of London Math Soc

View full text Add to dashboard Cite

show abstract

“…We have already shown that properties (P.1)-(P.5), and (P.7) hold true. The isoperimetric property is discussed (for Lipschitz sets) for monomial weights in [2,7,39,40,61], and for radial weights in [6,65,66]. The Cheeger problem in the monomial setting has also been considered in [7,26].…”

Section: Monomial and Radial Weightsmentioning

confidence: 99%

“…p (ω, L n ( • )) = {u ∈ L p (ω, L n ( • )) : |∇ X u| ∈ L p (ω, L n ( • ))}, where |∇ X u| = k i=1 (X i u)2 . In particular, a sub-Riemannian version of Green's identity ensures that, for an admissible bounded Ω and u ∈ W 1,2 (Ω) with Dirichlet boundary conditions on Ω, we have Ω |∇ X u| 2 dx dy = − Ω u∆ X u dx dy,…”

mentioning

confidence: 99%

The Cheeger problem in abstract measure spaces

Franceschi¹,

Pinamonti²,

Saracco³

et al. 2022

Preprint

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We consider non-negative σ-finite measure spaces coupled with a proper functional P that plays the role of a perimeter. We introduce the Cheeger problem in this framework and extend many classical results on the Cheeger constant and on Cheeger sets to this setting, requiring minimal assumptions on the pair measure space-perimeter. Throughout the paper, the measure space will never be asked to be metric, at most topological, and this requires the introduction of a suitable notion of Sobolev spaces, induced by the coarea formula with the given perimeter.

show abstract

On the Existence and Nonexistence of Isoperimetric Inequalities with Different Monomial Weights

Cited by 2 publications

References 27 publications

The Cheeger problem in abstract measure spaces

The Cheeger problem in abstract measure spaces

The Cheeger problem in abstract measure spaces

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