On Hilbert, Riemann, Neumann and Poincare problems for plane quasiregular mappings

TL;DR

This paper introduces a new approach to boundary value problems for quasiregular mappings, extending classical results to tangential limits and demonstrating infinite-dimensional solution spaces in multiply connected domains.

Contribution

It develops a novel method for analyzing tangential limits in quasiregular mappings, expanding the understanding of boundary behaviors in complex domains.

Findings

Solutions have infinite dimension for prescribed Jordan arc families.

New results on tangential limits in multiply connected domains.

Applications to Dirichlet, Riemann, Neumann, and Poincare problems for A-harmonic functions.

Abstract

Recall that the Hilbert (Riemann-Hilbert) boundary value problem for the Beltrami equations was recently solved for general settings in terms of nontangential limits and principal asymptotic values. Here it is developed a new approach making possible to obtain new results on tangential limits in multiply connected domains. It is shown that the spaces of the found solutions have the infinite dimension for prescribed families of Jordan arcs terminating in almost every boundary point. We give also applications of results obtained by us for the Beltrami equations to the boundary value problems of Dirichlet, Riemann, Neumann and Poincare for A-harmonic functions in the plane.