On the Hilbert boundary-value problem for Beltrami equations with singularities

Abstract

We investigate the Hilbert boundary-value problem for Beltrami equations \( \overline{\partial}f \) = μ∂f with singularities in generalized quasidisks D whose Jordan boundary ∂D consists of a countable collection of open quasiconformal arcs and, maybe, a countable collection of points. Such generalized quasicircles can be nowhere even locally rectifiable but include, for instance, all piecewise smooth curves, as well as all piecewise Lipschitz Jordan curves.Generally speaking, generalized quasidisks do not satisfy the standard (A)–condition in PDE by Ladyzhenskaya–Ural’tseva, in particular, the outer cone touching condition, as well as the quasihyperbolic boundary condition by Gehring–Martio that we assumed in our last paper for the uniformly elliptic Beltrami equations.In essence, here, we admit any countable collection of singularities of the Beltrami equations on the boundary and arbitrary singularities inside the domain D of a general nature. As usual, a point in \( \overline{D} \) is called a singularity of the Beltrami equation, if the dilatation quotient Kμ := (1 + |μ|)/(1 – |μ|) is not essentially bounded in all its neighborhoods.Presupposing that the coefficients of the problem are arbitrary functions of countable bounded variation and the boundary data are arbitrary measurable with respect to the logarithmic capacity, we prove the existence of regular solutions of the Hilbert boundary-value problem. As a consequence, we derive the existence of nonclassical solutions of the Dirichlet, Neumann, and Poincaré boundary-value problems for equations of mathematical physics with singularities in anisotropic and inhomogeneous media.