Dirichlet and Neumann boundary value problems for bi-polyanalytic functions on the bicylinder

Yanyan Cui; Chaojun Wang; Yanyan Cui; Chaojun Wang

doi:10.3934/math.2025220

AIMS Mathematics

2025, Volume 10, Issue 3: 4792-4818. doi: 10.3934/math.2025220

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Dirichlet and Neumann boundary value problems for bi-polyanalytic functions on the bicylinder

Yanyan Cui ^,,
Chaojun Wang

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

Received: 01 November 2024 Revised: 16 January 2025 Accepted: 11 February 2025 Published: 05 March 2025
MSC : 30C45, 32A30

Applying the Cauchy-Pompeiu formula and the properties of the singular integral operators on the unit disc, the specific representation of the solutions to the boundary value problems with the Dirichlet boundary conditions for bi-polyanalytic functions are obtained on the bicylinder. Also, the mixed-type boundary value problems of higher order for bi-polyanalytic functions were investigated. In addition, a system of complex partial differential equations with respect to polyanalytic functions with Neumann boundary conditions was discussed. On this foundation, the solutions to Neumann boundary value problems for bi-polyanalytic functions on the bicylinder were obtained. These results provide a favorable method for discussing other boundary value problems of bi-polyanalytic functions and the related systems of inhomogeneous complex partial differential equations of higher order in spaces of several complex variables.
- bi-polyanalytic functions,
- Dirichlet problems,
- Neumann problems,
- boundary value problems,
- complex partial differential equations
Citation: Yanyan Cui, Chaojun Wang. Dirichlet and Neumann boundary value problems for bi-polyanalytic functions on the bicylinder[J]. AIMS Mathematics, 2025, 10(3): 4792-4818. doi: 10.3934/math.2025220

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Abstract

Applying the Cauchy-Pompeiu formula and the properties of the singular integral operators on the unit disc, the specific representation of the solutions to the boundary value problems with the Dirichlet boundary conditions for bi-polyanalytic functions are obtained on the bicylinder. Also, the mixed-type boundary value problems of higher order for bi-polyanalytic functions were investigated. In addition, a system of complex partial differential equations with respect to polyanalytic functions with Neumann boundary conditions was discussed. On this foundation, the solutions to Neumann boundary value problems for bi-polyanalytic functions on the bicylinder were obtained. These results provide a favorable method for discussing other boundary value problems of bi-polyanalytic functions and the related systems of inhomogeneous complex partial differential equations of higher order in spaces of several complex variables.

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