The Dirichlet problem for Beltrami equations (Q2901677)

In the present paper a series of criteria for the existence of regular solutions of the Dirichlet problem for degenerate Beltrami equation is obtained. The research method is closely related to the so-called ring \(Q\)-homeomorphisms. Theorem 2 states that every regular homeomorphism in the class \(W_{\text{loc}}^{1,1}\) in the unit disk \({\mathbb D}\) is a ring \(Q\)-mapping at every boundary point \(z_0\in {\mathbb D}\), where \(Q\) is equal to the so-called tangential dilatation \(K_{\mu}^T(z)\) of \(f\). The main results of the work are the following. Suppose that \(\mu: {\mathbb D}\rightarrow {\mathbb D}\) is a measurable function and the corresponding maximal dilatation \(K_{\mu}(z)\) belongs to the class BMO or has finite mean oscillation in the closure of \({\mathbb D}\). Then, for every continuous function \(\varphi:\partial {\mathbb D}\rightarrow {\mathbb R}\), the Dirichlet problem NEWLINE\[NEWLINE\lim\limits_{z\rightarrow \zeta }\mathrm{Re }f(z)=\varphi(\zeta)\quad \text{for all } \zeta\in \partial {\mathbb D}NEWLINE\]NEWLINE for the Beltrami equation \(f_{\overline{z}}=\mu(z)f_z\) has a regular solution. Another result states the existence of the above problem under the condition of integral divergence \(\int\limits_{0}^{\delta(z_0)}\frac{dr}{\| K_{\mu}\|_{1}(r)}=\infty\) for all \(z_0\in \overline{{\mathbb D}}\) under some \(\delta(z_0)>0\). Some examples are also given in the final part of the paper.