Petkov I. The boundary behavior of solutions and the Dirichlet problem for the Beltrami equations

It is obtained a series of theorems on continuous and homeomorphic extension to the boundary of homeomorphic generalized solutions of the degenerate Beltrami equations. It is about the point-wise correspondence between the boundaries in the case of regular domains, and about the correspondence in the prime ends by Caratheodory in the case of irregular domains. On this basis, it is derived a series of theorems on the existence of regular (continuous, discrete and open) generalized solutions of the Dirichlet problem with continuous (and more general) boundary data in arbitrary Jordan domains, and also pseudoregular (with pole type singularities) as well as multivalent solutions in domains bounded by a finite collection of Jordan curves. In the case of domains with irregular boundaries, the corresponding theorems are formulated in terms of prime ends by Caratheodory.