A Density Result for Homogeneous Sobolev Spaces on Planar Domains

Abstract

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.

A density result on Orlicz-Sobolev spaces in the plane

“…In this section we prove Theorem 1.1, using the results of Section 2 and the partition of union from [28] that was recalled in Section 3. The polynomial approximation is exactly the same as in [28]. What is different is the way the estimates are carried out using Poincaré inequalities.…”

“…Such decomposition is standard in analysis, see for instance Whitney [34] or the book of Stein [32,Chapter VI]. We will use a version of the decomposition that was used in [28].…”

“…In this section we recall the decomposition of the domain Ω and the associated partition of unity that was obtained in [28]. In order to make the comparison between this paper and [28] easy, we use here the notation from [28].…”

“…The sets Hi are expanded by taking a connected component H i of a 2 −n−3 neighbourhood of Hi . The main property of the decomposition is that these expanded sets still satisfy H i ∩ H j = ∅ if and only if |i − j| ≤ 1 in a cyclical manner (Lemma 3.4 in [28]). Moreover, since the neighbourhoods are taken in the Euclidean distance, we may use an Euclidean partition of unity.…”

“…The validity of the approximation is proven using a Ψ − Ψ Poincaré inequality. The form of the polynomial approximation we use here was introduced in [28], where a density result was shown for homogeneous Sobolev spaces on simply connected planar domains. This was then extended to Gromov hyperbolic domains in higher dimensions in [27].…”