On the Dirichlet problem for beltrami equations with sources in simply connected domains

Abstract

In this paper, we present our recent results on the solvability of the Dirichlet problem Reω(z) → φ (ζ) as z → ζ, z∈ D, ζ∈ ∂D, with continuous boundary data φ: ∂D ???? R for degenerate Beltrami equations ωz =µ(z)ω2 + σ(z), |µ(z) ˂ 1 a.e., with sources σ: D → C that belong to the class Lp (D), p ˃ 2, and have compact supports in D. In the case of locally uniform ellipticity of the equations, we formulate, in arbitrary simply connected domains D of the complex plane C a series of eff ective integral criteria of the type of BMO, FMO, Calderon-Zygmund, Lehto and Orlicz on singularities of the equations at the boundary for existence of locally Hölder continuous solutions in the class W1.2loc (D)  of the Dirichlet problem with their representation through the so-called generalized analytic functions with sources.