# Kovtonyuk D. A.

### 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 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.

### Asymptotic behavior of generalized quasiisometries at a point

Kovtonyuk D. A., Salimov R. R.

Ukr. Mat. Zh. - 2011. - 63, № 4. - pp. 481-488

We consider $Q$-homeomorphisms with respect to the $p$-modulus. An estimate for a measure of a ball image is obtained under such mappings and the asymptotic behavior at zero is investigated.

### On the theory of hyper-$Q$-homeomorphisms

Ukr. Mat. Zh. - 2010. - 62, № 1. - pp. 139–144

We show that if a homeomorphism $f$ of a domain $D ⊂ R^n,\; n ≥ 2$, is a hyper-$Q$-homeomorphism with $Q ∈ L_{\text{loc}^1$ , then $f ∈ ACL$. As a consequence, this homeomorphism has partial derivatives and an approximation differential almost everywhere.