A Federer-style characterization of sets of finite perimeter on metric spaces

Lahti, Panu

doi:10.1007/s00526-017-1242-5

Cited by 18 publications

(22 citation statements)

References 34 publications

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“…Since V is open we have V ⊂ b 1 V ; recall (2.10) and the comment after it. We know that [19,Corollary 3.5], so in conclusion…”

Section: -Strict Subsetsmentioning

confidence: 83%

A new Cartan-type property and strict quasicoverings when p = 1 in metric spaces

Lahti¹

2018

Ann. Acad. Sci. Fenn. Math.

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In a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality, we prove a new Cartan-type property for the fine topology in the case p = 1. Then we use this property to prove the existence of 1-finely open strict subsets and strict quasicoverings of 1-finely open sets. As an application, we study fine Newton-Sobolev spaces in the case p = 1, that is, Newton-Sobolev spaces defined on 1-finely open sets. * 2010 Mathematics Subject Classification: 30L99, 31E05, 26B30.

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“…Since V is open we have V ⊂ b 1 V ; recall (2.10) and the comment after it. We know that [19,Corollary 3.5], so in conclusion…”

Section: -Strict Subsetsmentioning

confidence: 83%

A new Cartan-type property and strict quasicoverings when p = 1 in metric spaces

Lahti¹

2018

Ann. Acad. Sci. Fenn. Math.

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“…Perhaps the most important contribution of the current paper lies in our careful analysis of the 1-fine topology and the closely related notion of quasiopen sets. These have recently proved to be very useful concepts (see especially [34]) and we expect that a solid understanding of their properties will contribute to future research as well.…”

Section: Introductionmentioning

confidence: 99%

Capacities and 1-strict subsets in metric spaces

Lahti

2020

Nonlinear Analysis

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In a complete metric space that is equipped with a doubling measure and supports a Poincaré inequality, we study strict subsets, i.e. sets whose variational capacity with respect to a larger reference set is finite. Relying on the concept of fine topology, we give a characterization of those strict subsets that are also sets of finite perimeter, and then we apply this to the study of condensers as well as BV capacities. We also apply the theory to prove a pointwise approximation result for functions of bounded variation. Theorem 1.1. Let D ⊂ X and let E ⊂ X be a bounded set of finite perimeter with I E ⊂ D. Then cap 1 (I E , D) < ∞ if and only ifCap 1 (I E 1 \ fine-int D) = 0.Moreover, then cap 1 (I E , D) ≤ C a P (E, X) for a constant C a that depends only on the doubling constant of the measure and the constants in the Poincaré inequality. * 2010 Mathematics Subject Classification: 30L99, 31E05, 26B30.

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“…Much less is known (even in Euclidean spaces) in the case p = 1, but certain results of fine potential theory when p = 1 have been developed by the author in metric spaces in [28,29,30]. In quasiopen sets, the role of Lipschitz cutoff functions needs to be taken by Sobolev functions (often called Newton-Sobolev functions in metric spaces).…”

Section: Introductionmentioning

confidence: 99%

Discrete convolutions of $$\mathrm {BV}$$ functions in quasiopen sets in metric spaces

Lahti

2020

Calc. Var.

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We study fine potential theory and in particular partitions of unity in quasiopen sets in the case p = 1. Using these, we develop an analog of the discrete convolution technique in quasiopen (instead of open) sets. We apply this technique to show that every function of bounded variation (BV function) can be approximated in the BV and L ∞ norms by BV functions whose jump sets are of finite Hausdorff measure. Our results seem to be new even in Euclidean spaces but we work in a more general complete metric space that is equipped with a doubling measure and supports a Poincaré inequality. * 2010 Mathematics Subject Classification: 30L99, 31E05, 26B30.

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A Federer-style characterization of sets of finite perimeter on metric spaces

Cited by 18 publications

References 34 publications

A new Cartan-type property and strict quasicoverings when p = 1 in metric spaces

A new Cartan-type property and strict quasicoverings when p = 1 in metric spaces

Capacities and 1-strict subsets in metric spaces

Discrete convolutions of $$\mathrm {BV}$$ functions in quasiopen sets in metric spaces

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