Infinitesimal Geometry of Quasiconformal and Bi-Lipschitz Mappings in the Plane

Bojarski, Bogdan; Gutlyanskiî, Vladimir; Мартио, Олли; Ryazanov, Vladimir

doi:10.4171/122

Cited by 91 publications

(45 citation statements)

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“…It is necessary to mention also that the theory of the boundary behavior of Sobolev's mappings has significant applications to the boundary value problems for the Beltrami equations and for analogs of the Laplace equation in anisotropic and inhomogeneous media, see e.g. [2], [7]- [10], [12], [13], [19], [22], [24] and relevant references therein. For basic definitions and notations, discussions and historic comments in the mapping theory on the Riemann surfaces, see our previous papers [25]- [27].…”

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confidence: 99%

Prime ends in the Sobolev mapping theory on Riemann surfaces

Ryazanov¹,

Волков²

2017

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We prove criteria for continuous and homeomorphic extension to the boundary of mappings with finite distortion between domains on the Riemann surfaces by prime ends of Caratheodory.1. Introduction. The theory of the boundary behavior in the prime ends for the mappings with finite distortion has been developed in [11] for the plane domains and in [14] for the spatial domains. The pointwise boundary behavior of the mappings with finite distortion in regular domains on Riemann surfaces was recently studied by us in [26]. Moreover, the problem was investigated in regular domains on the Riemann manifolds for n ≥ 3 as well as in metric spaces, see e. ∈ (a, b). In what follows, |γ| denotes the locus of γ, i.e. the image γ ([a, b]).We act similarly to Caratheodory ([4]) under the definition of the prime ends of domains on a Riemann surface S, see Chapter 9 in [5]. First of all, recall that a continuous mapping σ : (ii) σ m splits D into 2 domains one of which contains σ m+1 and another one σ m−1 for every m > 1;2010 Mathematics Subject Classification: 30C62.

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confidence: 99%

Prime ends in the Sobolev mapping theory on Riemann surfaces

Ryazanov¹,

Волков²

2017

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“…In order to apply Theorem 4.2, we have to verify that J ω (z) ≡ 1 and |ω(z)| = |z| for z ∈ D. Indeed, noting that Re k = |k| 2 and substituting k(t) given by formula (5.12) into (5.15) we see that |ω(z)| = |z|, z ∈ D. The later implies that which plays important role in the study of different problems of contemporary analysis, see, e.g., [7], §13.2, [13], [14]. This function ω maps the unit disk D onto itself and transforms radial lines into spirals, infinitely winding about the origin, and it is just the volume preserving.…”

Section: Some Applicationsmentioning

confidence: 96%

“…In We start with the following statement, see, e.g., [7], p. 82. Analyzing formula (5.10), we arive at the following statement.…”

Section: Some Applicationsmentioning

confidence: 99%

Quasiconformal Mappings in the Theory of Semi-linear Equations

Gutlyanskiî

Ryazanov

2018

Lobachevskii J Math

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In this paper we study the semilinear partial differential equations in the plane, the linear part of which is written in a divergence form. The main result is given as a factorization theorem. This theorem states that every weak solution of such an equation can be represented as a composition of a weak solution of the corresponding isotropic equation in a canonical domain and a quasiconformal mapping agreed with a matrix-valued measurable coefficient appearing in the divergence part of the equation. The latter makes it possible, in particular, to remove the regularity restrictions on the boundary in the study of boundary value problems for such semilinear equations.

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“…In the paper we apply the length-area method originated in 1920's and isoperimetric inequality; see, e.g. [1], [4], [10], [17].…”

Section: Introductionmentioning

confidence: 99%