# Ryazanov V. I.

### Normality of the Orlicz - Sobolev classes

Ryazanov V. I., Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2016. - 68, № 1. - pp. 106-116

We establish a series of new criteria of equicontinuity and, hence, normality of the mappings of Orlicz – Sobolev classes in terms of inner dilatations.

### On the Theory of Prime Ends for Space Mappings

Kovtonyuk D. A., Ryazanov V. I.

Ukr. Mat. Zh. - 2015. - 67, № 4. - pp. 467-479

We present a canonical representation of prime ends in regular domains and, on this basis, study the boundary behavior of the so-called lower *Q*-homeomorphisms obtained as a natural generalization of quasiconformal mappings. We establish a series of effective conditions imposed on a function *Q*(*x*) for the homeomorphic extension of given mappings with respect to prime ends in domains with regular boundaries. The developed theory is applicable, in particular, to mappings of the Orlicz–Sobolev classes and also to finitely bi-Lipschitz mappings, which can be regarded as a significant generalization of the well-known classes of isometric and quasiisometric mappings.

### On the Orlicz–Sobolev Classes and Mappings with Bounded Dirichlet Integral

Ryazanov V. I., Salimov R. R., Sevost'yanov E. A.

Ukr. Mat. Zh. - 2013. - 65, № 9. - pp. 1254–1265

It is shown that homeomorphisms *f* in \( {{\mathbb{R}}^n} \) , *n* ≥ 2, with finite Iwaniec distortion of the Orlicz–Sobolev classes *W* ^{1,φ} _{loc} under the Calderon condition on the function φ and, in particular, the Sobolev classes *W* ^{1,φ} _{loc}, *p* > *n* - 1, are differentiable almost everywhere and have the Luzin (*N*) -property on almost all hyperplanes. This enables us to prove that the corresponding inverse homeomorphisms belong to the class of mappings with bounded Dirichlet integral and establish the equicontinuity and normality of the families of inverse mappings.

### On the Dirichlet problem for the Beltrami equations in finitely connected domains

Kovtonyuk D. A., Petkov I. V., Ryazanov V. I.

Ukr. Mat. Zh. - 2012. - 64, № 7. - pp. 932-944

We establish a series of criteria for the existence of regular solutions of the Dirichlet problem for degenerate Beltrami equations in arbitrary Jordan domains. We also formulate the corresponding criteria for the existence of pseudoregular and multivalued solutions of the Dirichlet problem in the case of finitely connected domains.

### On the boundary behavior of solutions of the Beltrami equations

Kovtonyuk D. A., Petkov I. V., Ryazanov V. I.

Ukr. Mat. Zh. - 2011. - 63, № 8. - pp. 1078-1091

We show that every homeomorphic solution of the Beltrami equation $\overline{\partial} f = \mu \partial f$ in the Sobolev class $W^{1, 1}_{\text{loc}}$ is a so-called lower $Q$-homeomorphism with $Q(z) = K_{\mu}(z)$, where $K_{\mu}$ is a dilatation quotient of this equation. On this basis, we develop the theory of the boundary behavior and the removability of singularities of these solutions.

### On the theory of the Beltrami equation

Ryazanov V. I., Srebro U., Yakubov E.

Ukr. Mat. Zh. - 2006. - 58, № 11. - pp. 1571–1583

We study ring homeomorphisms and, on this basis, obtain a series of theorems on the existence of the so-called ring solutions for degenerate Beltrami equations. A general statement on the existence of solutions for the Beltrami equations that extends earlier results is formulated. In particular, we give new existence criteria for homeomorphic solutions $f$ of the class $W^{1, 1}_{\text{loc}}$ with f −1 ∈ $f^{—1} \in W^{1, 2}_{\text{loc}}$ in terms of tangential dilatations and functions of finite mean oscillation. The ring solutions also satisfy additional capacity inequalities.

### On the noncompactness of classes of mappings with restrictions on dilation in measure

Potemkin V. L., Ryazanov V. I.

Ukr. Mat. Zh. - 1998. - 50, № 11. - pp. 1522–1531

We establish that the classes of mappings with restrictions in measure cannot be compact except for certain degenerate cases. In particular, this implies that any David class ΓI is noncompact.

### On compactness and metrizability of nuclear spaces of manifolds

Ukr. Mat. Zh. - 1998. - 50, № 6. - pp. 830–837

We prove a theorem on the sequential compactness of the so-called nuclear space of an arbitrary topological manifold.

### Criteria of convergence for quasiconformal mappings and their generalizations

Ukr. Mat. Zh. - 1996. - 48, № 5. - pp. 665-674

We establish necessary and sufficient conditions for the convergence of normalized homeomorphisms of Sobolev class in terms of the Fourier transforms of complex characteristics in the case where the upper bound of dilations is exponentially bounded in measure. This allows us to construct various metrics generating locally uniform convergence of mappings.

### Quasicqnformal mappings with restrictions in measure

Ukr. Mat. Zh. - 1993. - 45, № 7. - pp. 1009–1019

The principal result of the paper is a criterion of compactness for mappings quasiconformal in the mean. The semicontinuity of a deformation of homeomorphisms from the Sobolev class is also proved.

### Closure of classes of quasiconformal mappings with integral constraints

Ukr. Mat. Zh. - 1991. - 43, № 4. - pp. 435-440

### Convergence of characteristics of quasiconformal maps

Ukr. Mat. Zh. - 1986. - 38, № 2. - pp. 200–204