Quasiconformally homogeneous domains

Gehring, F. W.; Palka, Bruce

doi:10.1007/bf02786713

Cited by 332 publications

(217 citation statements)

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“…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.…”

Section: Asymptotics Of Solutionsmentioning

confidence: 99%

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

Valeriola

Schaftingen

2013

Arch Rational Mech Anal

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Section: Asymptotics Of Solutionsmentioning

confidence: 99%

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

Valeriola

Schaftingen

2013

Arch Rational Mech Anal

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“…This metric was introduced by Gehring and Palka in [7]. A curve γ joining x to y for which k Ω − length(γ) = k Ω (x, y) is called a quasihyperbolic geodesic.…”

Section: Preliminary Results On the Quasihyperbolic Metricmentioning

confidence: 99%

Quasihyperbolic boundary conditions and capacity: Hölder continuity of quasiconformal mappings

Koskela

Onninen

Tyson

2001

Comment. Math. Helv.

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Abstract. We prove that quasiconformal maps onto domains which satisfy a quasihyperbolic boundary condition are globally Hölder continuous in the internal metric. The primary improvement here over existing results along these lines is that no assumptions are made on the source domain. We reduce the problem to the verification of a capacity estimate in domains satisfing a quasihyperbolic boundary condition, which we establish using a combination of a chaining argument involving the Poincaré inequality on Whitney cubes together with Frostman's theorem.We also discuss related results where the quasihyperbolic boundary condition is slightly weakened; in this case the Hölder continuity of quasiconformal maps is replaced by uniform continuity with a modulus of continuity which we calculate explicitly. Mathematics Subject Classification (2000). Primary 26B35; Secondary 30C65, 30F45, 31B15, 46E35.

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“…This metric was introduced by Gehring and Palka in [6]. A curve γ joining x to y for which k Ω -length(γ) = k Ω (x, y) is called a quasihyperbolic geodesic.…”

Section: Notation and Definitionsmentioning

confidence: 99%