A higher-order Riemann-Hilbert problem for inhomogeneous complex partial differential equations

Yanyan Cui; Chaojun Wang; Yanyan Cui; Chaojun Wang

doi:10.3934/math.2026062

AIMS Mathematics

2026, Volume 11, Issue 1: 1463-1488. doi: 10.3934/math.2026062

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Research article

A higher-order Riemann-Hilbert problem for inhomogeneous complex partial differential equations

Yanyan Cui ^,,
Chaojun Wang

College of Mathematics and Statistics, Zhoukou Normal University, Zhoukou 466001, Henan, China

Received: 23 October 2025 Revised: 14 December 2025 Accepted: 19 December 2025 Published: 19 January 2026
MSC : 32A30, 30C45

In this paper, we mainly discussed an inhomogeneous Riemann-Hilbert boundary value problem for complex partial differential operators of higher order on the bicylinder $ D^2 $ in $ \mathbb{C}^2 $. Applying the classical Cauchy-Pompeiu formula and the Gauss theorem, we found out the solvable condition and obtained the specific solution of the inhomogeneous boundary value problem on the bicylinder $ D^2 $. The conclusion lays a solid foundation for further research on other types of boundary value problems concerning complex partial differential equations, such as mixed boundary value problems.
- boundary value problems,
- Cauchy-Pompeiu formula,
- Riemann-Hilbert problems,
- complex partial differential equations,
- Cauchy-Pompeiu formula
Citation: Yanyan Cui, Chaojun Wang. A higher-order Riemann-Hilbert problem for inhomogeneous complex partial differential equations[J]. AIMS Mathematics, 2026, 11(1): 1463-1488. doi: 10.3934/math.2026062

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Abstract

In this paper, we mainly discussed an inhomogeneous Riemann-Hilbert boundary value problem for complex partial differential operators of higher order on the bicylinder $ D^2 $ in $ \mathbb{C}^2 $. Applying the classical Cauchy-Pompeiu formula and the Gauss theorem, we found out the solvable condition and obtained the specific solution of the inhomogeneous boundary value problem on the bicylinder $ D^2 $. The conclusion lays a solid foundation for further research on other types of boundary value problems concerning complex partial differential equations, such as mixed boundary value problems.

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