Several boundary value problems arising from the extensions of analytic functions

Chaojun Wang; Yanyan Cui; Chaojun Wang; Yanyan Cui

doi:10.3934/math.2026563

AIMS Mathematics

2026, Volume 11, Issue 5: 13660-13682. doi: 10.3934/math.2026563

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Research article

Several boundary value problems arising from the extensions of analytic functions

Chaojun Wang ,
Yanyan Cui ^,

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

Received: 27 January 2026 Revised: 07 April 2026 Accepted: 21 April 2026 Published: 15 May 2026
MSC : 30C45, 32A30

In this paper, with the properties of $ \beta $-Cauchy type integrals and the Plemelj formula for the corresponding singular integrals, the Hilbert boundary value problems for $ \beta $-analytic functions were first discussed on the unit disc in $ \mathbb{C} $, and the concrete representations of the solutions were obtained. Thereafter, on the basis of the results for the Riemann boundary value problems for analytic functions, several Riemann problems for higher-order complex partial differential systems and $ (\lambda, 1) $ bi-analytic functions were investigated for the bicylinder in $ \mathbb{C}^2 $. The solutions to the problems and the corresponding solvable conditions were obtained.
- complex partial differential equations,
- Hilbert problems,
- Riemann problems,
- analytic functions,
- Cauchy type integral
Citation: Chaojun Wang, Yanyan Cui. Several boundary value problems arising from the extensions of analytic functions[J]. AIMS Mathematics, 2026, 11(5): 13660-13682. doi: 10.3934/math.2026563

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Abstract

In this paper, with the properties of $ \beta $-Cauchy type integrals and the Plemelj formula for the corresponding singular integrals, the Hilbert boundary value problems for $ \beta $-analytic functions were first discussed on the unit disc in $ \mathbb{C} $, and the concrete representations of the solutions were obtained. Thereafter, on the basis of the results for the Riemann boundary value problems for analytic functions, several Riemann problems for higher-order complex partial differential systems and $ (\lambda, 1) $ bi-analytic functions were investigated for the bicylinder in $ \mathbb{C}^2 $. The solutions to the problems and the corresponding solvable conditions were obtained.

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