Mappings of finite distortion: Formation of cusps III

@article{Koskela2007MappingsOF,
  title={Mappings of finite distortion: Formation of cusps III},
  author={Pekka Koskela and Juha Takkinen},
  journal={Acta Mathematica Sinica, English Series},
  year={2007},
  volume={26},
  pages={817-824}
}
We give sharp integrability conditions on the distortion of a planar homeomorphism that maps a standard cusp onto the unit disk. 
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23 Citations

23 Citations

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References

SHOWING 1-10 OF 29 REFERENCES
Mappings of finite distortion: The sharp modulus of continuity
We establish an essentially sharp modulus of continuity for mappings of subexponentially integrable distortion.
Lectures on n-Dimensional Quasiconformal Mappings
The modulus of a curve family.- Quasiconformal mappings.- Background in real analysis.- The analytic properties of quasiconformal mappings.- Mapping problems.
Extremal Mappings of Finite Distortion
The theory of mappings of finite distortion has arisen out of a need to extend the ideas and applications of the classical theory of quasiconformal mappings to the degenerate elliptic setting where…
Mappings of finite distortion: Capacity and modulus inequalities
Abstract We establish capacity and modulus inequalities for mappings of finite distortion under minimal regularity assumptions.
Mappings of finite distortion: formation of cusps
In this paper we consider the extensions of quasiconformal mappings f : B → Ωs to the whole plane, when the domain Ωs is a domain with a cusp of degree s > 0 and thus not an quasidisc. While these…
On the degenerate Beltrami equation
We study the well-known Beltrami equation under the assumption that its measurable complex-valued coefficient μ(z) has the norm ∥μ∥∞ = 1. Sufficient conditions for the existence of a homeomorphic…
Regularity of the Inverse of a Planar Sobolev Homeomorphism
Let be a domain. Suppose that f ∈ W1,1loc(Ω,R2) is a homeomorphism such that Df(x) vanishes almost everywhere in the zero set of Jf. We show that f-1 ∈ W1,1loc(f(Ω),R2) and that Df−1(y) vanishes…
Geometric Function Theory and Non-linear Analysis
0. Introduction and Overview 1. Conformal Mappings 2. Stability of the Mobius Group 3. Sobolev Theory and Function Spaces 4. The Liouville Theorem 5. Mappings of Finite Distortion 6. Continuity 7.…
Elliptic Partial Differential Equations and Quasiconformal Mappings in the Plane (Pms-48)
This book explores the most recent developments in the theory of planar quasiconformal mappings with a particular focus on the interactions with partial differential equations and nonlinear analysis.…
Mappings of finite distortion: Monotonicity and continuity
We study mappings f = ( f1, ..., fn) : Ω → Rn in the Sobolev space W loc (Ω,R n), where Ω is a connected, open subset of Rn with n ≥ 2. Thus, for almost every x ∈ Ω, we can speak of the linear…
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