On Hilbert problem for Beltrami equations in quasihyperbolic domains

TL;DR

This paper investigates the Hilbert boundary value problem for Beltrami equations in quasihyperbolic domains, proving existence of solutions with specific boundary data and exploring applications to classical boundary value problems in complex media.

Contribution

It establishes the existence of solutions for Beltrami equations with countably bounded variation coefficients in quasihyperbolic domains, extending classical boundary value problem theory.

Findings

Solutions exist with boundary data measurable w.r.t. logarithmic capacity.

The space of solutions has infinite dimension.

Applications to Dirichlet, Neumann, and Poincaré problems in anisotropic media.

Abstract

It is studied the Hilbert boundary value problem for the nondegenerate Beltrami equations in domains $D$ of the complex plane $\mathbb C$ with the so--called quasihyperbolic boundary condition. It is proved the existence of solutions of this problem with coefficients of countably bounded variation and with arbitrary boundary data that are measurable with respect to logarithmic capacity. It is shown that the dimension of the spaces of such solutions is infinite. It is also given applications to the Dirichlet, Neumann and Poincare boundary value problems for analogs of the Laplace equation in anisotropic and inhomogeneous media.