Denis Kovtonyuk

Boundary behavior and the Dirichlet problem for Beltrami equations

We show that arbitrary homeomorphic solutions to the Beltrami equations with generalized derivatives satisfy certain moduli inequalities. On this basis, we develope the theory of the boundary behavior of such solutions and prove a series of criteria for the existence of regular, pseudoregular and multi-valued solutions for the Dirichlet problem to the Beltrami equations in arbitrary Jordan domains and in arbitrary finitely connected domains bounded by mutually disjoint Jordan curves, correspondingly.

On the boundary behaviour of solutions to the Beltrami equations

It is shown that each homeomorphic solution to the Beltrami equation is the so-called lower Q-homeomorphism with Q(z) = K μ(z) where K μ(z) is the dilatation quotient of this equation. The theory of the boundary behaviour and the removability of singularities of such solutions are developed on this basis.