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In this paper we analyze the Hilbert boundary-value problem of the theory ofanalytic functions for an $(N+1)$-connected circular domain. An exactseries-form solution has already been derived for the case of continuouscoefficients. Motivated by the study of the Hall effect in a multiply connectedplate we extend these results by examining the case of discontinuouscoefficients. The Hilbert problem maps into the Riemann-Hilbert problem forsymmetric piece-wise meromorphic functions invariant with respect to asymmetric Schottky group. The solution to this problem is derived in terms oftwo analogues of the Cauchy kernel, quasiautomorphic and quasimultiplicativekernels. The former kernel is known for any symmetry Schottky group. We provethe existence theorem for the second, quasimultiplicative, kernel for anySchottky group (its series representation is known for the first class groupsonly). We also show that the use of an automorphic kernel requires the solutionto the associated real analogue of the Jacobi inversion problem which can bebypassed if we employ the quasiautomorphic and quasimultiplicative kernels. Weapply this theory to a model steady-state problem on the motion of chargedelectrons in a plate with $N+1$ circular holes with electrodes and dielectricson the walls when the conductor is placed at right angle to the appliedmagnetic field.

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