ON HILBERT BOUNDARY VALUE PROBLEMFOR BELTRAMI EQUATION

摘要

We study the Hilbert boundary value problemfor the Beltrami equation in the Jordan domains satisfying thequasihyperbolic boundary condition by Gehring Martio, generallyspeaking, without (A)-condition by Ladyzhenskaya Ural'tseva thatwas standard for boundary value problems in the PDE theory. Assumingthat the coefficients of the problem are functions of countablebounded variation and the boundary data are measurable with respectto the logarithmic capacity, we prove the existence of thegeneralized regular solutions. As a consequence, we derive theexistence of nonclassical solutions of the Dirichlet, Neumann andPoincar boundary value problems for generalizations of the Laplaceequation in anisotropic and inhomogeneous media.