Vladimir Ryazanov

Mappings with finite length distortion

General properties of mappings of finite metric distortion and of finite length distortion are studied. Uniqueness, equicontinuity, boundary behavior and removability of singularities are obtained under minimal additional assumptions.

Strong Ring Solutions of Beltrami Equations

Ring homeomorphisms are studied and then applied to an extension of the known Lehto existence theorem for degenerate Beltrami equations. On the basis of this extension, we establish a series of new general integral and measure conditions on the complex coefficient for the existence of ACL homeomorphic solutions. These criteria imply many of the known existence theorems as well as new results.

BMO-quasiconformal mappings

Plane BMO-quasiconformal and BMO-quasiregular mappings are introduced, and their basic properties are studied. This includes distortion, existence, uniqueness, representation, integrability, convergence and removability theorems, the reflection principle, boundary behavior and mapping properties.

On strong solutions of the Beltrami equations

We formulate general principles on the existence of homeomorphic absolutely continuous on lines (ACL) solutions for the Beltrami equations with degeneration and derive from them a series of criteria and, in particular, a generalization and strengthening of the well-known Lehto existence theorem. Furthermore, we prove that in all these cases there exist the so-called strong ring solutions satisfying additional moduli conditions which play a great role in the research of various properties of such solutions.

Finite mean oscillation and the Beltrami equation

We prove the existence and uniqueness of homeomorphic ACL solutions to the Beltrami equation in the case when the dilatation coefficient of the equation has a majorant of finite mean oscillation.

Integral conditions in the theory of the Beltrami equations

It is shown that many recent and new results regarding the existence of ACL homeomorphic solutions for degenerate Beltrami equations with integral constraints follow from our extension of the well-known Lehto existence theorem.

On the Beltrami Equations with Two Characteristics

The Beltrami equations of the first type

Boundary behavior and the Dirichlet problem for Beltrami equations

f_{\bar{z}} = \mu(z)f_{z}