Boundary behavior of quasiconformal mappings in n-space

Näkki, Raimo

doi:10.5186/aasfm.1971.484

Cited by 34 publications

(43 citation statements)

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“…As in the theory of qusiconformal mappings [63,52,39], in order to extend a given map h ∈ E(X, Y) continuously to the closure of X, we must assume some geometric regularity of ∂X and ∂Y. Definition 1.3.…”

Section: Theorem 12 Under the Assumption Of Theorem 11 If Y Is Momentioning

confidence: 99%

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Deformations of finite conformal energy: Boundary behavior and limit theorems

Iwaniec

Onninen

2011

Trans. Amer. Math. Soc.

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Abstract. We study homeomorphisms h : X onto −→ Y between two bounded domains in R n having finite conformal energyWe consider the behavior of such mappings, including continuous extension to the closure of X and injectivity of h : X → Y. In general, passing to the weak W 1,n -limit of a sequence of homeomorphisms h j : X → Y one loses injectivity. However, if the mappings in question have uniformly bounded L 1 -average of the inner distortion, then, for sufficiently regular domains X and Y, their limit map h : X onto −→ Y is a homeomorphism. Moreover, the inverse map−→ X enjoys finite conformal energy and has integrable inner distortion as well.

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Section: Theorem 12 Under the Assumption Of Theorem 11 If Y Is Momentioning

confidence: 99%

“…We also emphasize that if X separates X 0 from two punctures X i = {a i } and X j = {a j }, then (52) diam(X, X) |a i − a j | .…”

Section: Estimates Of the Distance To The Boundarymentioning

confidence: 99%

Deformations of finite conformal energy: Boundary behavior and limit theorems

Iwaniec

Onninen

2011

Trans. Amer. Math. Soc.

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“…The notions of strong accessibility and weak flatness at boundary points of a domain in R n , as defined in [91], see also [92,94,123,153], are localizations and generalizations of the corresponding notions introduced in [121,122], compare with the properties P 1 and P 2 introduced by Väisälä in [182] and also with the quasiconformal accessibility and the quasiconformal flatness introduced by Näkki in [130]. Many theorems on homeomorphic extension to the boundary for quasiconformal mappings and their generalizations are valid under the condition of weak flatness for boundaries.…”

Section: Figurementioning

confidence: 99%

Toward the theory of Orlicz–Sobolev classes

Kovtonyuk

Ryazanov

Salimov

et al. 2014

St. Petersburg Math. J.

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It is shown that, under a Calderón type condition on the function ϕ, the continuous open mappings that belong to the Orlicz-Sobolev classes W 1,ϕ loc have total differential almost everywhere; this generalizes the well-known theorems of Gehring-Lehto-Menchoff in the case of R 2 and of Väisälä in R n , n ≥ 3. Appropriate examples show that the Calderón type condition is not only sufficient but also necessary. Moreover, under the same condition on ϕ, it is also proved that the continuous mappings of class W 1,ϕ loc and, in particular, of class W 1,p loc for p > n−1 have Lusin's (N )-property on a.e. hyperplane. On that basis, it is shown that, under the same condition on ϕ, the homeomorphisms f with finite distortion of class W 1,ϕ loc and, in particular, those belonging to W 1,p loc for p > n−1, are what is called lower Q-homeomorphisms, where Q is equal to their outer dilatation K f ; also, they are so-called ring Q * -homeomorphisms with Q * = K n−1 f . The latter fact makes it possible to fully apply the theory of the boundary and local behavior of the ring and lower Q-homeomorphisms, as developed earlier by the authors, to the study of mappings in the Orlicz-Sobolev classes. Part 1. Differentiability and behavior of Hausdorff 's measures in the Orlicz-Sobolev classes2010 Mathematics Subject Classification. Primary 46E35. Key words and phrases. Moduli of families of curves and surfaces, mappings with bounded and finite distortion, differentiability, Lusin and Sard properties, Sobolev classes, Orlicz-Sobolev classes, boundary and local behavior.

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“…with the properties P 1 and P 2 by Väisälä in [49] and also with the quasiconformal accessibility and the quasiconformal flatness by Näkki in [37]. Many theorems on a homeomorphic extension to the boundary of quasiconformal mappings and their generalizations are valid under the condition of weak flatness of boundaries.…”

Section: Weakly Flat and Strongly Accessible Boundariesmentioning

confidence: 99%

On boundary behavior of generalized quasi-isometries

Kovtonyuk

Ryazanov

2011

JAMA

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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.

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Boundary behavior of quasiconformal mappings in n-space

Cited by 34 publications

References 0 publications

Deformations of finite conformal energy: Boundary behavior and limit theorems

Deformations of finite conformal energy: Boundary behavior and limit theorems

Toward the theory of Orlicz–Sobolev classes

On boundary behavior of generalized quasi-isometries

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