On Hilbert and Riemann problems for generalized analytic functions and applications

Abstract

The research of the Dirichlet problem with arbitrary measurable data for harmonic functions is due to the famous dissertation of Luzin. The present paper is devoted to various theorems on the existence of nonclassical solutions of the Hilbert and Riemann boundary value problems with arbitrary measurable data for generalized analytic functions by Vekua and the corresponding applications to the Neumann and Poincare problems for generalized harmonic functions. Our approach is based on the geometric (theoretic-functional) interpretation of boundary values in comparison with the classical operator approach in PDE. First of all, here it is proved theorems on the existence of solutions to the Hilbert boundary value problem with arbitrary measurable data for generalized analytic functions in arbitrary Jordan domains with rectifiable boundaries in terms of the natural parameter and angular (nontangential) limits, moreover, in arbitrary Jordan domains in terms of harmonic measure and principal asymptotic values. Moreover, it is established the existence theorems on solutions for the appropriate boundary value problems of Hilbert and Riemann with arbitrary measurable data along the so-called Bagemihl–Seidel systems of Jordan arcs terminating at the boundary in arbitrary domains whose boundaries consist of finite collections of rectifiable Jordan curves. On this basis, it is established the corresponding existence theorems for the Poincare boundary value problem on the directional derivatives and, in particular, for the Neumann problem with arbitrary measurable data to the Poisson equations.