Potential Theory on Sierpiński Carpets

Ntalampekos, Dimitrios

doi:10.1007/978-3-030-50805-0

Cited by 4 publications

(3 citation statements)

References 22 publications

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“…A proof of the first two parts of this lemma can be found in [20]. In particular, part (i) is proved in [20, Remark 2.3.5] and part (ii) is proved in [20, Lemma 2.3.4].…”

Section: Preliminaries On Curves Of Bounded Curvaturementioning

confidence: 99%

On the Hausdorff dimension of the residual set of a packing by smooth curves

Maio

Ntalampekos

2022

Journal of London Math Soc

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“…A proof of the first two parts of this lemma can be found in [20]. In particular, part (i) is proved in [20, Remark 2.3.5] and part (ii) is proved in [20, Lemma 2.3.4].…”

Section: Preliminaries On Curves Of Bounded Curvaturementioning

confidence: 99%

On the Hausdorff dimension of the residual set of a packing by smooth curves

Maio

Ntalampekos

2022

Journal of London Math Soc

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“…In addition to its importance in classical uniformization problems, this method has played a central role in recent developments on the uniformization of fractal metric spaces, cf. [1], [3], [4], [8], [17].…”

Section: The Corresponding Quotient Map Is πmentioning

confidence: 99%

Uniformization of planar domains by exhaustion

Rajala¹

2021

Preprint

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We study the method of finding conformal maps onto circle domains by approximating with finitely connected subdomains. Every domain D ⊂ Ĉ admits exhaustions, i.e., increasing sequences of finitely connected subdomains Dj whose union is D. By Koebe's theorem, each Dj admits a conformal map fD j from Dj onto a circle domain fD j (Dj ). Assuming fD j → f , our goal is to find out if f (D) is also a circle domain.We present a countably connected D with an exhaustion (Dj ) so that (fD j ) has a limit whose image is not a circle domain, and a domain Ω with an exhaustion (Ωj ) so that (fΩ j ) has a limit whose image has uncountably many non-point complementary components.On the other hand, we prove that every exhaustion (Dj) of a countably connected D admits a refinement so that the image of the corresponding limit map is a circle domain. Our result extends the He-Schramm theorem on the uniformization of countably connected domains and provides a new proof.

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“…The main motivation for Theorem 1.6 was to study regularity properties of a certain type of infimizers that appear in the setting of Sierpiński carpets and are called carpet-harmonic functions; see [34,Chapter 2]. Namely, these infimizers are restrictions of monotone Sobolev functions (under some geometric assumptions) and the approximation Theorem 1.6 implies some absolute continuity properties for these functions.…”

Section: Introductionmentioning

confidence: 99%

Monotone Sobolev Functions in Planar Domains: Level Sets and Smooth Approximation

Ntalampekos

2020

Arch Rational Mech Anal

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We prove that almost every level set of a Sobolev function in a planar domain consists of points, Jordan curves, or homeomorphic copies of an interval. For monotone Sobolev functions in the plane we have the stronger conclusion that almost every level set is an embedded 1-dimensional topological submanifold of the plane. Here monotonicity is in the sense of Lebesgue: the maximum and minimum of the function in an open set are attained at the boundary. Our result is an analog of Sard's theorem, which asserts that for a C 2 -smooth function in a planar domain almost every value is a regular value. As an application, using the theory of p-harmonic functions, we show that monotone Sobolev functions in planar domains can be approximated uniformly and in the Sobolev norm by smooth monotone functions.

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Potential Theory on Sierpiński Carpets

Cited by 4 publications

References 22 publications

On the Hausdorff dimension of the residual set of a packing by smooth curves

On the Hausdorff dimension of the residual set of a packing by smooth curves

Uniformization of planar domains by exhaustion

Monotone Sobolev Functions in Planar Domains: Level Sets and Smooth Approximation

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