Quasiextremal distance domains and extension of quasiconformal mappings

“…Theorem 12.6 in [30], and QED-domains have weakly flat boundaries, the following consequence of Theorem 4.1 is a far-reaching generalization of the Gehring-Martio theorem on a homeomorphic extension to boundaries of quasiconformal mappings between QED-domains, cf. [13] and [35]. …”

Section: The Main Resultsmentioning

confidence: 99%

“…It is known that every uniform domain is a QED-domain, but there are QEDdomains that are not uniform, see [13]. Bounded convex domains and bounded domains with smooth boundaries are simple examples of uniform domains and, consequently, QED-domains as well as domains with weakly flat boundaries.…”

Section: Weakly Flat and Strongly Accessible Boundariesmentioning

confidence: 99%

On boundary behavior of generalized quasi-isometries

Kovtonyuk

Ryazanov

2011

JAMA

View full text Add to dashboard Cite

It is established a series of criteria for continuous and homeomorphic extension to the boundary of the so-called lower Q-homeomorphisms f between domains in R n = R n ∪ {∞}, n 2, under integral constraints of the type Φ(Q n−1 (x)) dm(x) < ∞ with a convex non-decreasing functionIt is shown that integral conditions on the function Φ found by us are not only sufficient but also necessary for a continuous extension of f to the boundary. It is given also applications of the obtained results to the mappings with finite area distortion and, in particular, to finitely bi-Lipschitz mappings that are a far reaching generalization of isometries as well as quasiisometries in R n . In particular, it is obtained a generalization and strengthening of the well-known theorem by Gehring-Martio on a homeomorphic extension to boundaries of quasiconformal mappings between QED (quasiextremal distance) domains.2000 Mathematics Subject Classification: Primary 30C65; Secondary 30C75 Key words: mappings with finite area distortion, moduli of families of surfaces, finitely biLipschitz mappings, weakly flat and strongly accessible boundaries.

show abstract

“…It is known that every uniform domain is a QED-domain but there exist QED-domains that are not uniform, see [8]. Bounded convex domains and bounded domains with smooth boundaries are simple examples of uniform domains and, consequently, QED-domains, as well as domains with weakly flat boundaries.…”

Section: (∆(E F ; D)) P (23)mentioning

confidence: 99%