Boundary value problems for the Beltrami
equations are due to the famous Riemann dissertation (1851) in the
simplest case of analytic functions and to the known works of
Hilbert (1904, 1924) and Poincare (1910) for the corresponding
Cauchy--Riemann system. Of course, the Dirichlet problem was well
studied for uniformly elliptic systems, see, e.g., \cite{Boj} and
\cite{Vekua}. Moreover, the corresponding results on the Dirichlet
problem for degenerate Beltrami equations in the unit disk can be
found in the monograph \cite{GRSY}.
In our article \cite{KPR1}, see also \cite{KPR3} and \cite{KPR5}, it
was shown that each generalized homeomorphic solution of a Beltrami
equation is the so-called lower $Q-$homeomorphism with its
dilatation quotient as $Q$ and developed on this basis the theory of
the boundary behavior of such solutions. In the next papers
\cite{KPR2} and \cite{KPR4}, the latter made possible us to solve
the Dirichlet problem with continuous boundary data for a wide
circle of degenerate Beltrami equations in finitely connected Jordan
domains, see also [\citen{KPR5}--\citen{KPR7}].
Similar problems were also investigated in the case of bounded
finitely connected domains in terms of prime ends by Caratheodory in
the papers [\citen{KPR9}--\citen{KPR10}] and [\citen{P1}--\citen{P2}].
Finally, in the present paper, we prove a series of effective
criteria for the existence of pseudo\-re\-gu\-lar and multi-valued
solutions of the Dirichlet problem for the degenerate Beltrami
equations in arbitrary bounded finitely connected domains in terms
of prime ends by Caratheodory.
The BMO-Dirichlet problem for elliptic systems in the upper half-space and quantitative characterizations of VMO
