Modulus of curve families and extremality of spiral-stretch maps

Proceedings of the London Mathematical Society, 2005

arXiv (Cornell University), 2022

For an arbitrary convex function Ψ : [1, ∞) → [1, ∞), we consider uniqueness in the following two related extremal problems: Problem A (boundary value problem): Establish the existence of, and describe the mapping f , achieving inf f D Ψ(K(z, f)) dz : f :D →D a homeomorphism in W 1,1 0 (D) + f 0. Here the data f 0 :D →D is a homeomorphism of finite distortion with D Ψ(K(z, f 0)) dz < ∞-a barrier. Next, given two homeomorphic Riemann surfaces R and S and data f 0 : R → S a diffeomorphism.

Archive for Rational Mechanics and Analysis, 2010

We determine the extremal mappings with smallest mean distortion for mappings of annuli. As a corollary, we find that the Nitsche harmonic maps are Dirichlet energy minimizers among all homeomorphisms h : A(r, R) → A(r , R ). However, outside the Nitsche range of the modulus of the annuli, within the class of homeomorphisms, no such energy minimizers exist. In this case we identify the BV-limits of minimizers.

Mathematical Research Letters, 2005

Reports of the National Academy of Sciences of Ukraine

The present paper is a natural continuation of our previous papers [1-3], where the reader can find the corresponding historic comments and a discussion of many definitions and relevant results. The given papers were devoted to the theory of the boundary behavior of mappings with finite distortion by Iwaniec. Here, we will develop the theory of the boundary behavior of the so-called mappings with finite length distortion first introduced in [4] for n  , 2 n , see also Chapter 8 in [5]. As was shown in [6], such mappings, generally speaking, are not mappings with finite distortion by Iwaniec, because their first partial derivatives can be not locally integrable. At the same time, this class is a natural generalization of the well-known classes of bi-Lipschitz mappings, as well as isometries and quasi-isometries.

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