  It is developed the theory of the boundary behavior of homeomorphic solutions
of the Beltrami equations ${\bar{\partial}}f=\mu\,{\partial}f$ of the Sobolev
class $W^{1,1}_{\rm loc}$ with respect to prime ends of domains. On this basis,
under certain conditions on the complex coefficient ${\mu}$, it is proved the
existence of regular solutions of its Dirichlet problem in arbitrary simply
connected domains and pseudoregular as well as multivalent solutions in
arbitrary finitely connected domains with continuous boundary data in terms of
prime ends.

Complex Variables