A Density Problem for Sobolev Spaces on Planar Domains

Koskela, Pekka; Zhang, Yi Ru-Ya

doi:10.1007/s00205-016-0994-y

Cited by 10 publications

(25 citation statements)

References 8 publications

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“…The argument for the Jordan domain case is similar to the proof of [11,Corollary 1.2]. Recall that for any two non-empty subsets X and Y of R n , the…”

Section: 3mentioning

confidence: 85%

“…After this we decompose a bounded domain Ω (which is δ-Gromov hyperbolic with respect to the quasihyperbolic metric) into several parts via Lemma 2.1, and then construct a corresponding partition of unity. In [11] conformal mappings and planar geometry were applied to obtain the desired composition. In our setting, we cannot rely on mappings nor on simple geometry.…”

Section: Definition 12mentioning

confidence: 99%

“…More recently, in [8] it was shown by Giacomini and Trebeschi that, for bounded simply connected planar domains, W 1, 2 (Ω) is dense in W 1, p (Ω) for all 1 ≤ p < 2. Motivated by the results above, Koskela and Zhang proved in [11] that for any bounded simply connected domain and any 1 ≤ p < ∞, W 1, ∞ (Ω) is dense in W 1, p (Ω), and C ∞ (R 2 ) is dense in W 1, p (Ω) when Ω is Jordan.…”

Section: Introductionmentioning

confidence: 99%

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A density problem for Sobolev spaces on Gromov hyperbolic domains

Koskela

Rajala

Zhang

2017

Nonlinear Analysis: Theory, Methods & Applications

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“…The argument for the Jordan domain case is similar to the proof of [11,Corollary 1.2]. Recall that for any two non-empty subsets X and Y of R n , the…”

Section: 3mentioning

confidence: 85%

Section: Definition 12mentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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A density problem for Sobolev spaces on Gromov hyperbolic domains

Koskela

Rajala

Zhang

2017

Nonlinear Analysis: Theory, Methods & Applications

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“…Let us fix one of the component A . Inside the topological rectangle A , via an argument similar to [ 14 , Sect. 3.2], we can find two hyperbolic geodesics and joining to such that and where the constants are absolute.…”

Section: Prerequisitesmentioning

confidence: 99%

Duality of capacities and Sobolev extendability in the plane

Zhang

2021

Complex Anal Synerg

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We reveal relations between the duality of capacities and the duality between Sobolev extendability of Jordan domains in the plane, and explain how to read the curve conditions involved in the Sobolev extendability of Jordan domains via the duality of capacities. Finally as an application, we give an alternative proof of the necessary condition for a Jordan planar domain to be $$W^{1,\,q}$$ W 1 , q -extension domain when $$2<q<\infty$$ 2 < q < ∞ .

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“…This method of inner extension for this purpose has been used in [13], [14] and [20]. In [13] this extension is done using the conformal parametrization of simply connected domains. In [20], the topology of the plane is utilized directly to construct an approximating sequence although the hyperbolic structure of planar simply connected domains plays an implicit role in the construction.…”

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confidence: 99%

A Density Result for Homogeneous Sobolev Spaces on Planar Domains

2018

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We show that in a bounded Gromov hyperbolic domain Ω smooth functions with bounded derivatives C ∞ (Ω) ∩ W k,∞ (Ω) are dense in the homogeneous Sobolev spaces L k,p (Ω). IntroductionWe continue the study of density of functions with bounded derivatives in the space of Sobolev functions in a domain in R n . It was shown by Koskela-Zhang [13] that for a simply connected planar domain Ω ⊂ R 2 , W 1,∞ is dense in W 1,p and in the special case of Jordan domains also C ∞ (R 2 ) ∩ W 1,∞ (Ω) is dense. The above result of Koskela-Zhang has been generalized to have the density of W k,∞ in the homogeneous Sobolev space L k,p for k ∈ N in planar simply connected domains by Nandi-Rajala-Schultz [20]. In dimensions higher than two however simply connectedness is not sufficient (see for example [14]). Recall that simply connected planar domains are negatively curved in the (quasi) hyperbolic metric. A useful metric generalization of negatively curved spaces was introduced by Gromov [8], in the context of group theory. Following Bonk-Heinonen-Koskela [4], we call a domain Gromov hyperbolic if, when equipped with the quasihyperbolic metric, it is δ-hyperbolic in the sense of Gromov, for some δ ≥ 0 (see Section 3 for definitions). Gromov hyperbolicity has turned out to be a sufficient condition for the density of W 1,∞ in W 1,p , as shown by Koskela-Rajala-Zhang [14], and these are primarily the domains we consider in this paper.Let us mention here a few similarities between our setting and the planar simply connected case. We recall that simply connected domains with the quasihyperbolic metric (equivalent to the hyperbolic metric by the Koebe distortion theorem) in the plane are Gromov hyperbolic and that the quasihyperbolic geodesics are unique (see Luiro [17]); conversely, a Gromov hyperbolic domain with uniqueness of quasihyperbolic geodesics is simply connected. The latter is of course true in higher dimensions as well (indeed, in this case, for any pair of points, any curve γ joining the points is homotopic to the unique quasihyperbolic geodesic Γ joining the given points, with the homotopy given by quasihyperbolic geodesics joining the points γ(t) and Γ(t) once Γ is parametrized suitably). Gromov hyperbolic domains are often seen as a topological generalization of planar simply connected domains. It was shown by Bonk-Heinonen-Koskela [4] that Gromov hyperbolic domains are conformally equivalent to suitable uniform metric spaces equipped with the quasihyperbolic metric and corresponding suitable measures. Finally, we note that in higher dimensions, simply connectedness alone does not imply Gromov hyperbolicity; consider for example the unit ball in R 3 deformed to have a cuspidal-wedge along an equator.

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A Density Problem for Sobolev Spaces on Planar Domains

Abstract: Abstract. We prove that for a bounded simply connected domain Ω ⊂ R 2 , the Sobolev space W 1, ∞ (Ω) is dense in W 1, p (Ω) for any 1 ≤ p < ∞. Moreover, we show that if Ω is Jordan, then C ∞ (R 2 ) is dense in W 1, p (Ω) for 1 ≤ p < ∞.

Cited by 10 publications

References 8 publications

A density problem for Sobolev spaces on Gromov hyperbolic domains

A density problem for Sobolev spaces on Gromov hyperbolic domains

Duality of capacities and Sobolev extendability in the plane

A Density Result for Homogeneous Sobolev Spaces on Planar Domains

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