Uniform domains and the quasi-hyperbolic metric

Desingularization of Vortex Rings and Shallow Water Vortices by a Semilinear Elliptic Problem

“…When Ω is a uniform domain, this is equivalent with the uniform continuity with respect to the quasi-hyperbolic metric on Ω. Those metrics are equivalent to the Poincaré metric on the ball and on the half-plane [26,27,33]. Assumption (A 5 ) is satisfied under the assumptions of proposition 2.3 or of lemma 2.7.…”

Weighted Trudinger-type inequalities

“…For any pair of points x, y, there always exists a quasihyperbolic geodesic, i.e. a path γ x,y such that k Ω (x, y) = k Ω (γ x,y ); see [18]. Clearly, 1 + k Ω (γ) is greater than some fixed factor times the number of Whitney cubes through which γ passes.…”

Weak Bloch Functions, ∅-Uniform and ∅-John Domains

“…John domains arise naturally in distortion problems of conformal and quasiconformal mappings [3], [4], [5], [7]. An inner uniform domain is a domain intermediate between a uniform domain and a John domain.…”

“…We have some important bounds for quasihyperbolic metric and the bounds involve metrics j D and j D (see [1], [2], [5], [9], [10], [11]). We define…”

“…These bounds may be reversed exactly if the domain is uniform (or inner uniform) as follows([2, Theorem 5.3.5], [5], [9], [11]). …”