Riemann–Hilbert problems for null-solutions to iterated generalized Cauchy–Riemann equation on upper half ball

Abstract

We study Riemann–Hilbert boundary value problems with variable coefficients for axially symmetric null-solutions to the iterated generalized Cauchy–Riemann equation, defined over the upper half unit ball centred at the origin in four-dimensional Euclidean space. First, we prove an Almansi-type decomposition theorem for axially symmetric null-solutions to the iterated generalized Cauchy–Riemann equation. Then, we give integral representation solutions to the Riemann–Hilbert problems for axially symmetric null-solutions to iterated generalized Cauchy–Riemann equation over the upper half unit ball centred at the origin in four-dimensional Euclidean space. In particular, we derive solutions to the Schwarz problem for axially symmetric null-solutions to iterated generalized Cauchy–Riemann equation over the upper half unit ball centred at the origin in four-dimensional Euclidean space. Finally, we further extend the results to axially symmetric null-solutions to over the upper half unit ball centred at the origin in four-dimensional Euclidean space.