Open Discrete Mappings Research Articles

On modulus inequality of the order $p$ for the inner dilatation

The article is devoted to mappings with boundedand finite distortion of planar domains. Our investigations aredevoted to the connection between mappings of the Sobolev class andupper bounds for the distortion of the modulus of families of paths.For this class, we have proved the Poletsky-type inequality withrespect to the so-called inner dilatation of the order~$p.$ Weseparately considered the situations of homeomorphisms and mappingswith branch points. In particular, we have established thathomeomorphisms of the Sobolev class satisfy the upper estimate ofthe distortion of the modulus at the inner and boundary points ofthe domain. In addition, we have proved that similar estimates ofcapacity distortion occur at the inner points of the domain for opendiscrete mappings. Also, we have shown that open discrete and closedmappings satisfy some estimates of the distortion of the modulus offamilies of paths at the boundary points. The results of themanuscript are obtained mainly under the condition that theso-called inner dilatation of mappings is locally integrable. Themain approach used in the proofs is the choice of admissiblefunctions, using the relations between the modulus and capacity, andconnections between different modulus of families of paths (similarto Hesse, Ziemer and Shlyk equalities). In this context, we haveobtained some lower estimate of the modulus of families of paths inSobolev classes. The manuscript contains some examples related toapplications of obtained results to specific mappings.

On removal of isolates singularities of open discrete mappings

We study mappings with finite distortion in $\mathbb{R}^n$, $n \geqslant 2$. For the classes of mappings with branching that satisfy modal inequalities we obtain the set of theorems on removal of isolated singularities.

Boundary behaviour of open discrete mappings on Riemannian manifolds. II

The boundary behaviour of classes of ring mappings, which generalize quasiconformal mappings in the sense of Gehring, is under investigation. Theorems proving that they have continuous boundary extensions are established in terms of prime ends of regular domains. Results on the equicontinuity of mappings in these classes in the closure of a fixed domain are also established in these terms. Bibliography: 45 titles.

Boundary behaviour of open discrete mappings on Riemannian manifolds. II

Изучается граничное поведение классов кольцевых отображений на римановых многообразиях, являющихся обобщением квазиконформных отображений по Герингу. В терминах простых концов регулярных областей получены теоремы об их непрерывном продолжении на границу области. В этих же терминах доказаны утверждения о равностепенной непрерывности указанных классов в замыкании заданной области. Библиография: 45 названий. В опубликованной печатной версии статьи по техническим причинам в названии статьи пропущена цифра II.

On the Boundary Behavior of Some Classes of Mappings

We study the boundary behavior of closed open discrete mappings from the Sobolev and Orlicz–Sobolev classes in ℝn, n ≥ 3. It is proved that such a mapping f can be extended by continuity to a boundary point x0 ∈ ∂ D of a domain D ⊂ ℝn whenever its inner dilatation of order α > n− 1 has a majorant from the finite mean oscillation class at the point in question. Another sufficient condition for the existence of a continuous extension is the divergence of some integral. We also prove some results on the continuous extension of such a mapping to an isolated boundary point.

Boundary behaviour of open discrete mappings on Riemannian manifolds

The paper is concerned with problems of continuous extension of certain classes of mappings on Riemannian manifolds to boundary points of a given domain. In particular, the so-called ring mappings are shown to be continuously extendable to an isolated boundary point. Analogous theorems are also derived under more general conditions on the boundaries of the given and the target domains. As an application of the machinery thus developed, an arbitrary open discrete boundary-preserving mapping from the Orlicz-Sobolev class is shown to extend continuously to an isolated boundary point. Bibliography: 40 titles.

On the Equicontinuity of Mappings with Branching in the Closure of the Domain

We study the problem of local behavior of mappings f : $$ D\to \overline{{\mathbb{R}}^n} $$ , n ≥ 2, in $$ \overline{D} $$ . Under certain conditions imposed on a measurable function Q(x), Q: D→[0, ∞], and the boundaries of the domains D and D ′ = f(D), we show that a family of open discrete mappings f : $$ D\to \overline{{\mathbb{R}}^n} $$ with a characteristic of quasiconformality Q(x) is equicontinuous in $$ \overline{D} $$ .

On local behavior of mappings with unbounded characteristic

This article is a survey of the earlier results of the author. We study space mappings with branching that satisfy modulus inequalities. For classes of these mappings, we obtain several sufficient conditions for the normality of families. Moreover, we prove that the normal families of the so-called Q-mappings have the logarithmic order of growth in a neighborhood of a point. We obtain a result on the normal families of open discrete mappings f: D → C {a, b} from the class W loc 1,1 with finite distortion that do not take at least two fixed values a ≠ b in C whose maximal dilatation has a majorant of finite mean oscillation at every point. This result is an analog of the well-known Montel theorem for analytic functions.

On the Local Behavior of Open Discrete Mappings from the Orlicz–Sobolev Classes

The paper is devoted to the study of mappings with unbounded characteristic of quasiconformality and, in particular, of mappings with finite distortion, which are now extensively studied. We establish theorems on the equicontinuity of the families of mappings that belong to the Orlicz–Sobolev class for n ≥ 3 and have finite distortions. To do this, we also study some auxiliary classes of mappings. In fact, we analyze the relationship between the so-called lower Q-mappings and some capacity-type inequalities.

Absolute continuity on paths of spatial open discrete mappings

We prove that open discrete mappings of Sobolev classes \(W_\mathrm{loc}^{1, p},\)\(p>n-1,\) with locally integrable inner dilatations admit \(ACP_p^{\,-1}\)-property, which means that these mappings are absolutely continuous on almost all preimage paths with respect to p-module. In particular, our results extend the well-known Poletskiĭ lemma for quasiregular mappings. We also establish the upper bounds for p-module of such mappings in terms of integrals depending on the inner dilatations and arbitrary admissible functions.

Estimates for Jacobian and Dilatation Coefficients of Open Discrete Mappings with Controlled p-Module

We study open discrete mappings whose p-module of curve families is integrally restricted and establish various estimates for the Jacobian and dilatation coefficients. We also show that such mappings are close to Lipschitz mappings, quasiregular mappings and mappings of finite distortion. The sharpness of these estimates is illustrated by several examples.

On a Vӓisӓlӓ-type inequality for the angular dilatation of mappings and some of its applications

For a subtype of mappings with finite distortion f : D → D′, D, D′ ⊂ ℝ n ; n ≥ 2; which admit the existence of branch points, a modular inequality playing an essential role in the study of various problems of planar and spatial mappings is established. As an application, the problem of removal of isolated singularities of open discrete mappings with finite length distortion is investigated.

Open discrete mappings with unbounded coefficient of quasi-conformality on Riemannian manifolds

The paper is concerned with problems at the intersection of the theory of spatial quasi-conformal mappings and the theory of Riemann surfaces. Theorems on the local behaviour of one class of open discrete mappings with unbounded coefficient of quasi-conformality, which map between arbitrary Riemannian manifolds, are obtained. Such mappings are also shown to extend to isolated points of the boundary of the domain. Some results on the local behaviour of Sobolev and Orlicz-Sobolev classes are obtained as an application. Bibliography: 52 titles.

Some properties of open, discrete, generalized ring mappings

We study properties of open, discrete ring mappings satisfying generalized modular inequalities. Such mappings generalize the known class of quasiregular mappings and their extensions known as mappings of finite distortion. We apply our results to open discrete ring mappings satisfying condition (N) and having local inverses, and we focus especially on the case . We show that such mappings cannot have essential singularities and also that Zoric’s theorem can hold in this case under some conditions even if . This is in contrast even with the known case of quasiregular mappings.

Analog of the Montel Theorem for Mappings of the Sobolev Class with Finite Distortion

We study the classes of mappings with unbounded characteristic of quasiconformality and obtain a result on the normal families of open discrete mappings f : D→ ℂ {a, b} from the class W loc 1,1 with finite distortion that do not take at least two fixed values a 6≠b in ℂ whose maximal dilatation has a majorant of finite mean oscillation at every point. This result is an analog of the well-known Montel theorem for analytic functions and is true, in particular, for the so-called Q-mappings.

On local properties of one class of mappings on Riemannian manifolds

The present paper is devoted to questions located at the junction of the theories of space quasiconformal mappings and Riemannian surfaces. Theorems of local behavior of a class of open discrete mappings with unbounded characteristic of quasiconformality on arbitrary Riemannian manifolds are obtained.

On the boundary behavior of open discrete mappings with unbounded characteristic

We study the problem of extension of mappings $ f:D\to \overline{{{{\mathbb{R}}^n}}} $ , n ≥ 2, to the boundary of a domain D. Under certain conditions imposed on a measurable function Q(x), Q:D → [0, ∞], and the boundaries of the domains D and D′ = f (D), we show that an open discrete mapping $ f:D\to \overline{{{{\mathbb{R}}^n}}} $ , n ≥ 2, with quasiconformality characteristic Q(x) can be extended to the boundary ∂D by continuity. The obtained statements extend the corresponding Srebro’s result to mappings with bounded distortion.

Analogs of the Ikoma–Schwartz lemma and Liouville theorem for mappings with unbounded characteristic

We obtain results on the local behavior of open discrete mappings $ f:D \to {\mathbb{R}^n} $ , n ≥ 2, that satisfy certain conditions related to the distortion of capacities of condensers. It is shown that, in an infinitesimal neighborhood of zero, the indicated mapping cannot grow faster than an integral of a special type that corresponds to the distortion of the capacity under this mapping, which is an analog of the well-known Ikoma growth estimate proved for quasiconformal mappings of the unit ball into itself and of the classic Schwartz lemma for analytic functions. For mappings of the indicated type, we also obtain an analog of the well-known Liouville theorem for analytic functions.

On some properties of generalized quasiisometries with unbounded characteristic

We consider a family of open discrete mappings $$ f:D \to \overline {{{\mathbb R}^n}} $$ that distort, in a special way, the p-modulus of a family of curves that connect the plates of a spherical condenser in a domain D in $$ {{\mathbb R}^n} $$ ; p > n-1; p < n; and bypass a set of positive p-capacity. We establish that this family is normal if a certain real-valued function that controls the considered distortion of the family of curves has finite mean oscillation at every point or only logarithmic singularities of order not higher than n - 1: We show that, under these conditions, an isolated singularity x 0 ∈ D of a mapping $$ f:D\backslash \left\{ {{x_0}} \right\} \to \overline {{{\mathbb R}^n}} $$ is removable, and, moreover, the extended mapping is open and discrete. As applications, we obtain analogs of the known Liouville and Sokhotskii–Weierstrass theorems.

Open discrete mappings having local ACL n inverses

We consider open, discrete mappings between domains from R n satisfying condition (N), having local ACL n inverses on D∖B f , so that μ n (B f ) = 0, H*(·, f) < ∞ on B f and . For this class of mappings (or even for larger classes of open, discrete mappings) we generalize the important modular inequality of Poleckii, also using the modular estimates of the spherical rings from Cristea (Local homeomorphism having local ACL n inverses, Complex Var. Elliptic Equ. 53(1) (2008), 77–99). We continue the work from the same paper by generalizing some basic facts from the theory of quasiregular mappings. We give equicontinuity results, Picard, Montel and Liouville type theorems, estimates of the modulus of continuity, analogues of Schwarz's lemma, eliminability results and boundary extensions theorems. Together with the multiple extensions of Zoric's theorem from the paper, we establish strong generalizations of one of the most important theorems from the theory of quasiregular mappings. We also extend similar results given in some recent classes of functions larger than the class of quasiregular functions, as the class of mappings of finite distortion and satisfying condition (𝒜), or the class of mappings of finite dilatation with dilatation in the bounded mean oscillation class.