Riemann–Hilbert problems for null-solutions to iterated generalized Cauchy–Riemann equations in axially symmetric domains. (May 2016) - He, Fuli; Ku, Min; Kähler, Uwe; Sommen, Frank; Bernstein, Swanhild
Abstract: 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 α k ϕ = 0, k ⩾ 2 ( k ∈ N ), α ∈ R .