On Hilbert, Poincare and Riemann problems for Beltrami equations with sources

First of all, we study the Hilbert boundary value problem for the Beltrami equations with sources in Jordan domains of the complex plane. Assuming that the coefficient of the problem is a function of countable bounded variation and the boundary date is measurable with respect to the logarithmic capacity, we prove the existence of nonclassical solutions of the problem in the sense of limits along all non-tangential paths quasi-everywhere on the boundary.

In another case, when the notion of the boundary limits is weakened and is understood in the sense of limits along arbitrary systems of arcs by Bagemihl–Seidel, the Hilbert problem is already solved by us with arbitrary measurable coefficients. In this case, we also prove here similar solvability theorems for the Riemann boundary value problem on the conjugation, including also nonlinear boundary conditions.

Our final theorems on mixed boundary conditions allow us, in particular, to obtain a solution to the Poincare boundary value problem on directional derivatives, including the nonlinear case. Note that all found solutions are represented through generalized analytic functions with sources.