Riemann-Hilbert problems for null-solutions to iterated generalized Cauchy-Riemann equations in axially symmetric domains

摘要

We consider Riemann-Hilbert boundary value problems (for short RHBVPs) with variable coefficients for axially symmetric poly-monogenic functions, i.e., null-solutions to iterated generalized Cauchy-Riemann equations, defined in axially symmetric domains. This extends our recent results about RHBVPs with variable coefficients for axially symmetric monogenic functions defined in four-dimensional axially symmetric domains. First, we construct the Almansi-type decomposition theorems for poly-monogenic functions of axial type. Then, making full use of them, we give the integral representation solutions to the RHBVP considered. As a special case, we derive solutions to the corresponding Schwarz problem. Finally, we generalize the result obtained to functions of axial type which are null-solutions to perturbed iterated generalized Cauchy-Riemann equations D-alpha(k)phi = 0, k >= 2(k is an element of N), alpha is an element of R. (C) 2016 Elsevier Ltd. All rights reserved.