On the Dirichlet problem for degenerate Beltrami equations

We study the Dirichlet problem \(\lim\limits_{z \to \zeta} \text{Re } f(z) = \varphi(\zeta)\) as \(z \to \zeta\), \(z \in D\), \(\zeta \in \partial D\), with continuous boundary data \(\varphi(\zeta)\) in arbitrary simply connected bounded domains \(D\) of the complex plane \(\mathbb{C}\), where \(f\) satisfies the degenerate Beltrami equation \(f_{\bar{z}} = \mu(z) f_z\), \(|\mu(z)| < 1\), a. e. in \(D\). We give 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.