Modulus of curve families and extremality of spiral-stretch maps

2010

It is shown that every homeomorphism f of finite distortion in the plane is the so-called lower Q-homeomorphism with Q(z) = K f (z), and, on this base, it is developed the theory of the boundary behavior of such homeomorphisms.

2015

In the present paper, it was studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec, also to solutions of the Beltrami equations, as well as to finitely bi--Lipschitz mappings that a far--reaching extension of the known classes of isometric and quasiisometric mappings.

Lobachevskii Journal of Mathematics

In the present paper, it was studied the boundary behavior of the so-called lower Q-homeomorphisms in the plane that are a natural generalization of the quasiconformal mappings. In particular, it was found a series of effective conditions on the function Q(z) for a homeomorphic extension of the given mappings to the boundary by prime ends. The developed theory is applied to mappings with finite distortion by Iwaniec, also to solutions of the Beltrami equations, as well as to finitely bi-Lipschitz mappings that a far-reaching extension of the known classes of isometric and quasiisometric mappings.

Transactions of the American Mathematical Society, 2003

We establish an essentially sharp modulus of continuity for mappings of subexponentially integrable distortion.

Journal d'Analyse Mathématique, 2008

In this paper, we define the notion of asymptotic spirallikeness (a generalization of asymptotic starlikeness) in the Euclidean space C n . We consider the connection between this notion and univalent subordination chains. We introduce the notions of A-asymptotic spirallikeness and A-parametric representation, where A ∈ L(C n , C n ), and prove that if ∞ 0 e (A−2m(A)In )t dt < ∞ (this integral is convergent if k + (A) < 2m(A)), then a mapping f ∈ S(B n ) is Aasymptotically spirallike if and only if f has A-parametric representation, i.e., if and only if there exists a univalent subordination chain f (z, t) such that Df (0, t) = e At , {e −At f (·, t)} t≥0 is a normal family on B n and f = f (·, 0). In particular, a spirallike mapping with respect to A ∈ L(C n , C n ) with ∞ 0 e (A−2m(A)In )t dt < ∞ has A-parametric representation. We also prove that if f is a spirallike mapping with respect to an operator A such that A + A * = 2In, then f has parametric representation (i.e., with respect to the identity). Finally, we obtain some examples of asymptotically spirallike mappings. 267 268 I. GRAHAM, H. HAMADA, G. KOHR AND M. KOHR

Journal für die reine und angewandte Mathematik (Crelles Journal), 2000

Page 1. Mappings of finite distortion: Capacity and modulus inequalities Pekka Koskela ∗ Jani Onninen ∗ Abstract ... 02000 Mathematics Subject Classification: 30C65 1 Page 2. so-called path lifting can be applied to mappings that are both open and dis-crete. ...

Dynamical Systems, 2012

We give an example illustrating that two notions of bounded distortion for C 1 expanding maps in R are different.

Siberian Mathematical Journal

Journal d'Analyse Mathématique, 2004

General properties of mappings of finite metric distortion and of finite length distortion are studied. Uniqueness, equicontinuity, boundary behavior and removability of singularities are obtained under minimal additional assumptions.

Journal d'Analyse Mathématique, 2008

In all dimensions k = 1, ..., n − 1, we show that mappings f in R n with finite distortion of hyperarea satisfy certain modulus inequalities in terms of inner and outer dilatation of the mappings.

2003

We establish a radius of injectivity for locally homeomorphic mappings of finite distortion in space, under minimal integrability assumptions on the distortion.

has proved a theorem about local homeomorphism of mappings with bounded distortion with the coefficient of distortion close to identity. Independently, this result was established by O. Martio, C. Rickman and J. Väisälä . In the present paper, we use ideas of V. M. Goldshtein, Yu. G. Reshetnyak, O. Martio, C. Rickman, J. Väisälä and prove a theorem about local homeomorphism provided that a nonconstant mapping with bounded distortion and the coefficient of distortion 1 on the Carnot group is a homeomorphism. The problem is that the structure of 1-quasiregular mappings is unknown on an arbitrary Carnot group. It is known ] that 1-quasiregular mappings on the one-point compactification of the Heisenberg group H n are generated by left translations, dilations, actions of a group element in SU (1, n), and the inversion The problem is still open for an arbitrary Carnot group. 1991 Mathematics Subject Classification. 22E30, 30C65, 43A80.

Annales Academiae Scientiarum Fennicae Mathematica, 2013

We propose a method by modulus of curve families to identify extremal quasiconformal mappings in the Heisenberg group. This approach allows to study minimizers not only for the maximal distortion but also for a mean distortion functional, where the candidate for the extremal map is not required to have constant distortion. As a counterpart of a classical Euclidean extremal problem, we consider the class of quasiconformal mappings between two spherical annuli in the Heisenberg group. Using logarithmic-type coordinates we can define an analog of the classical Euclidean radial stretch map and discuss its extremal properties both with respect to the maximal and the mean distortion. We prove that our stretch map is a minimizer for a mean distortion functional and it minimizes the maximal distortion within the smaller subclass of sphere-preserving mappings.

Inventiones Mathematicae, 2001

Annales de l'Institut Henri Poincare (C) Non Linear Analysis, 2010

Let f : Ω → R 2 be a mapping of finite distortion, where Ω ⊂ R 2. Assume that the distortion function K(x, f) satisfies e K(•,f) ∈ L p loc (Ω) for some p > 0. We establish optimal regularity and area distortion estimates for f. Especially, we prove that |Df | 2 log β−1 (e + |Df |) ∈ L 1 loc (Ω) for every β < p. This answers positively well known conjectures due to Iwaniec and Martin [13] and to Iwaniec, Koskela and Martin [14].