Hilbert problem for a multiply connected circular domain and the analysis of the Hall effect in a plate

TL;DR

This paper extends the analysis of the Hilbert boundary-value problem for multiply connected circular domains to include discontinuous coefficients, with applications to modeling the Hall effect in a plate with multiple holes.

Contribution

It introduces a new solution approach using quasiautomorphic and quasimultiplicative kernels, bypassing the Jacobi inversion problem for Schottky groups.

Findings

Derived an exact series-form solution for discontinuous coefficients.

Proved the existence of the quasimultiplicative kernel for any Schottky group.

Applied the theory to a model of electron motion in a multiply connected plate.

Abstract

In this paper we analyze the Hilbert boundary-value problem of the theory of analytic functions for an $(N+1)$-connected circular domain. An exact series-form solution has already been derived for the case of continuous coefficients. Motivated by the study of the Hall effect in a multiply connected plate we extend these results by examining the case of discontinuous coefficients. The Hilbert problem maps into the Riemann-Hilbert problem for symmetric piece-wise meromorphic functions invariant with respect to a symmetric Schottky group. The solution to this problem is derived in terms of two analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicative kernels. The former kernel is known for any symmetry Schottky group. We prove the existence theorem for the second, quasimultiplicative, kernel for any Schottky group (its series representation is known for the first class groups only). We also show that the use of an automorphic kernel requires the solution to the associated real analogue of the Jacobi inversion problem which can be bypassed if we employ the quasiautomorphic and quasimultiplicative kernels. We apply this theory to a model steady-state problem on the motion of charged electrons in a plate with $N+1$ circular holes with electrodes and dielectrics on the walls when the conductor is placed at right angle to the applied magnetic field.