Riemann–Hilbert problems for null-solutions to iterated generalized Cauchy–Riemann equation on upper half ball. Issue 11 (1st November 2020)

• Record Type: Journal Article
• Authors: Ku, Min; He, Fuli; He, Xiuli
• Is Part Of: Complex variables and elliptic equations. Volume 65:Issue 11(2020)
• Page Start: 1902
• Page End: 1918
• Publication Date: 2020-11-01
• Subjects: Riemann–Hilbert problems -- iterated generalized Cauchy–Riemann operator -- quaternion analysis -- variable coefficients -- axial symmetry; Primary: 30E25 -- 35Q15 -- 31A25 -- 31B20 -- Secondary: 31B10 -- 35J56 -- 35J58; Functions of complex variables -- Periodicals; Differential equations, Elliptic -- Periodicals; 515.905
• DOI: 10.1080/17476933.2019.1664484
• Languages: English
• ISSNs: 1747-6933

Abstract

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 D α k φ = 0, k ≥ 2 ( k ∈ N ), α ∈ R over the upper half unit ball centred at the origin in four-dimensional Euclidean space.