Riemann–Hilbert Problems for Axially Symmetric Monogenic Functions in $${\mathbb {R}}^{n+1}$$

Abstract

We focus on the Clifford-algebra valued variable coefficients Riemann–Hilbert boundary value problems $$\big ($$ for short RHBVPs $$\big )$$ for axially monogenic functions on Euclidean space $${\mathbb {R}}^{n+1},n\in {\mathbb {N}}$$ . With the help of Vekua system, we first make one-to-one correspondence between the RHBVPs considered in axial domains and the RHBVPs of generalized analytic function on complex plane. Subsequently, we use it to solve the former problems, by obtaining the solutions and solvable conditions of the latter problems, so that we naturally get solutions to the corresponding Schwarz problems. In addition, we also use the above method to extend the case to RHBVPs for axially null-solutions to $$\big ({\mathcal {D}}-\alpha \big )\phi =0,\alpha \in {\mathbb {R}}$$ .