On the Theory of Prime Ends for Space Mappings

Abstract

It is given a canonical representation of prime ends in regular spatial domains and, on this basis, it is studied the boundary behavior of the so-called lower Q-homeomorphisms that are the natural generalization of the quasiconformal mappings. In particular, it is found a series of effective conditions on the function Q(x) for a homeomorphic extension of the given mappings to the boundary by prime ends in domains with regular boundaries. The developed theory is applied, in particular, to mappings of the classes of Sobolev and Orlicz-Sobolev and also to finitely bi-Lipschitz mappings that a farreaching extension of the well-known classes of isometric and quasiisometric mappings.

On boundary extension of mappings in metric spaces in the terms of prime ends

“…Analog of the following lemma was proved in [MRSY,Lemma 13.4] (see also [KR,Lemma 4] and [Sm,Lemma 5]).…”

On compact classes of solutions of Dirichlet problem in simply connected domains

“…where η is any Lebesgue measurable function satisfying (9) for r 1 → r m and r 2 → ε 0 , in addition, Q m := Q fm corresponds to the function Q in (8). Let us to prove the inequality…”

On boundary Hölder logarithmic continuity of mappings in some domains

“…Consider the following definition that has been proposed Näkki [13], cf. [14]. The boundary of a domain D is called locally quasiconformal, if every point x 0 ∈ ∂D has a neighborhood U, for which there exists a quasiconformal mapping ϕ of U onto the unit ball B n ⊂ R n such that ϕ(∂D ∩ U) is the intersection of the unit sphere B n with a coordinate hyperplane x n = 0, where x = (x 1 , .…”

“…The definition of a prime end used below may be found in [15], cf. [14]. We say that a bounded domain D in R n is regular, if D can be quasiconformally mapped to a domain with a locally quasiconformal boundary whose closure is a compact in R n , and, besides that, every prime end in D is regular.…”