BMO and Dirichlet problem for degenerate Beltrami equation

Following Bojarski and Vekua, we have studied the Dirichlet problem \(\lim\limits_{z\to\zeta}{\rm Re}\,f(z)=\varphi(\zeta)\) as \(z\to\zeta\), \(z\in D,\zeta\in \partial D\), with continuous boundary data \(\varphi(\zeta)\) in bounded domains \(D\) of the complex plane \(\mathbb C\), where \(f\) satisfies the degenerate Beltrami equation \(f_{\bar{z}}=\mu(z)f_z\), \(|\mu(z)| \lt 1\), a.e. in \(D\). Assuming that \(D\) is an arbitrary simply connected domain, we have established, in terms of \(\mu\), the BMO and FMO criteria, as well as a number of other integral criteria, on the existence and representation of regular discrete open solutions to the stated above problem. We have also proven similar theorems on the existence of multi-valued solutions to the problem with single-valued real parts in an arbitrary bounded domain \(D\) with no boundary component degenerated to a single point. Finally, we have given a similar solvability and representation results concerning the Dirichlet problem in such domains for the degenerate \(A\)-harmonic equation associated with the Beltrami equation.