On the stability of phi -uniform domains

📝 Abstract

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n $. In the sequel, we investigate a class of domains, so called $\varphi $-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\varphi$ from $[0,\infty)$ to itself. Finally, we discuss a number of stability properties of $\varphi $-uniform domains. In particular, we show that the class of $\varphi $-uniform domains is stable in the sense that removal of a geometric sequence of points from a $\varphi $-uniform domain yields a $\varphi_1 $-uniform domain.

💡 Analysis

We study two metrics, the quasihyperbolic metric and the distance ratio metric of a subdomain $G \subset {\mathbb R}^n $. In the sequel, we investigate a class of domains, so called $\varphi $-uniform domains, defined by the property that these two metrics are comparable with respect to a homeomorphism $\varphi$ from $[0,\infty)$ to itself. Finally, we discuss a number of stability properties of $\varphi $-uniform domains. In particular, we show that the class of $\varphi $-uniform domains is stable in the sense that removal of a geometric sequence of points from a $\varphi $-uniform domain yields a $\varphi_1 $-uniform domain.

📄 Content

For a subdomain G R n and x, y ∈ G the distance ratio metric j G is defined by

where δ G (x) denotes the Euclidean distance from x to ∂G. Sometimes we abbreviate δ G by writing just δ . The above form of the j G metric, introduced in [14], is obtained by a slight modification of a metric that was studied in [3,4]. The quasihyperbolic metric of G is defined by the quasihyperbolic length minimizing property

where Γ(x, y) represents the family of all rectifiable paths joining x and y in G, and ℓ k (γ) is the quasihyperbolic length of γ (cf. [4]). For a given pair of points x, y ∈ G, the infimum is always attained [3], i.e., there always exists a quasihyperbolic geodesic J G [x, y] which minimizes the above integral, k G (x, y) = ℓ k (J G [x, y]) and furthermore with the property that the distance is additive on the geodesic:

If the domain G is emphasized we call J G [x, y] a k G -geodesic. In this paper, sometimes we also use the terminology distance for the term metric.

The following well-known properties of the above two metrics are useful in this paper. (i) For x, y ∈ G R n , we have k G (x, y) ≥ j G (x, y) [4];

(ii) Monotonicity property: if G 1 and G 2 are domains, with G 2 ⊂ G 1 R n , then for all x, y ∈ G 2 we have k G 1 (x, y) ≤ k G 2 (x, y). It is obvious that this property holds for the distance ratio metric j G as well. In 1979, Martio and Sarvas introduced the class of uniform domains [10].

Definition 1.1. A domain D in R n is said to be c-uniform if there exists a constant c with the property that each pair of points z 1 , z 2 in D can be joined by a rectifiable arc γ in D satisfying (cf. [10,12])

(1) min j=1,2 ℓ(γ[z j , z]) ≤ c δ D (z) for all z ∈ γ, and

(2) ℓ(γ) ≤ c |z 1z 2 |, where ℓ(γ) denotes the arclength of γ, γ[z j , z] the part of γ between z j and z. Also, we say that γ is a uniform arc. A domain is said to be uniform if it is c-uniform for some constant c > 0.

In the same year, Gehring and Osgood [3] characterized uniform domains in terms of an upper bound for the quasihyperbolic metric as follows: a domain G is uniform if and only if there exists a constant C ≥ 1 such that (1.2) k G (x, y) ≤ Cj G (x, y) for all x, y ∈ G. As a matter of fact, the above inequality appeared in [3] in a form with an additive constant on the right hand side: it was shown by Vuorinen [14, 2.50] that the additive constant can be chosen to be 0. This observation leads to the definition of ϕ-uniform domains introduced in [14].

for all x, y ∈ G.

In the sequel, Väisälä has also investigated this class of domains [12] (see also [13] and references therein). He also pointed out that these two classes of domains are same provided ϕ is a slow function. We make sure that, in this paper, we use the terminology c-uniform for constants c, and frequently use ψ-uniform, ϑ-uniform and ϕ-uniform for functions ψ, ϑ, ϕ.

In Section 2, we introduce notation and preliminary results that we need in the latter sections. The structure of the rest of the sections covers mainly on ϕ-uniform domains.

In Section 3, we construct several examples of ϕ-uniform domains and compare with their complementary domains and with quasiconvex domains. We also prove that the image domain of a ϕ-uniform domain under bilipschitz mappings of R n is ψ-uniform, where ψ is depending on ϕ and the bilipschitz constant.

In Section 4, we present our main results (e.g. see Theorems 4.8 and 4.23) on ϕ-uniform domains in the following form:

where ϑ depends only on ϕ and ψ.

Note that one of our proofs involves a control function of a fixed parameter on which ϑ also depends. Idea behind this is to obtain various other stability properties of ϕ-uniform domains to use as preparatory results to prove the main theorems in the above type. In particular, it is shown that the class of ϕ-uniform domains is stable in the sense that removal of a geometric sequence of points from a ϕ-uniform domain leads to a ϕ 1 -uniform domain.

We shall now specify some necessary notation, definitions and facts that we frequently use in this paper. The standard unit vectors in the Euclidean n-space R n (n ≥ 2) are represented by e 1 , e 2 , . . . , e n . We write x ∈ R n as a vector (x 1 , x 2 , . . . , x n ). The Euclidean line segment joining points x and y is denoted by [x, y]. For x, y, z ∈ R n , the smallest angle at y between the vectors xy and zy is denoted by ∡(x, y, z). The one point compactification of R n (so-called the Möbius n-space) is defined by R n = R n ∪ {∞}. We denote by B n (x, r) and S n-1 (x, r), the Euclidean ball and sphere with radius r centered at x respectively. We set B n (r) := B n (0, r) and S n-1 (r) := S n-1 (0, r). Let G be a domain (open connected non-empty set) in R n . The boundary, closure and diameter of G are denoted by ∂G, G and diam G respectively. In what follows, all paths γ ⊂ G are required to be rectifiable, i.e. ℓ(γ) < ∞ where ℓ(γ) stands for the Euclidean length of γ. Given x, y ∈ G, Γ(x, y) stands for the collection of all rec

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