On Hilbert boundary value problem for Beltrami equation

Abstract

We study the Hilbert boundary value problem for the Beltrami equation in the Jordan domains satisfying the quasihyperbolic boundary condition by Gehring–Martio, generally speaking, without (A)-condition by Ladyzhenskaya–Ural’tseva that was standard for boundary value problems in the PDE theory. Assuming that the coefficients of the problem are functions of countable bounded variation and the boundary data are measurable with respect to the logarithmic capacity, we prove the existence of the generalized regular solutions. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann and Poincaré boundary value problems for generalizations of the Laplace equation in anisotropic and inhomogeneous media.