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April 2022 Integral representations in the complex plane and iterated boundary value problems
Mohamed Akel, Heinrich Begehr, Alip Mohammed
Rocky Mountain J. Math. 52(2): 381-413 (April 2022). DOI: 10.1216/rmj.2022.52.381


Fundamental solutions to differential operators lead to integral operators providing integral representation formulas for solutions to related differential equations. Proper modifications of the fundamental solutions result in integral operators which are related to certain boundary value problems. For complex partial differential operators of arbitrary order in the plane, fundamental solutions are achievable by properly integrating the Cauchy kernel. Particular such complex model differential operators are the polyanalytic and the polyharmonic operators. A hierarchy of integral operators is available for these model operators leading to polyanalytic Cauchy–Schwarz and to polyharmonic Green, Neumann, Robin, and hybrid Green integral operators. The theory is supplemented here by constructing a new polyanalytic Pompeiu (Pompeiu–Vekua) integral operator of any order adjusted to (iterated) Neumann boundary conditions.


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Mohamed Akel. Heinrich Begehr. Alip Mohammed. "Integral representations in the complex plane and iterated boundary value problems." Rocky Mountain J. Math. 52 (2) 381 - 413, April 2022.


Received: 12 November 2020; Revised: 16 November 2020; Accepted: 2 July 2021; Published: April 2022
First available in Project Euclid: 17 May 2022

Digital Object Identifier: 10.1216/rmj.2022.52.381

Primary: 30E25 , 31A10
Secondary: 30G20 , 31A30 , 35J40

Keywords: Cauchy–Schwarz–Pompeiu representation , Dirichlet , hybrid Green functions , Neumann , polyanalytic , polyanalytic Neumann problem in domains with harmonic Green function , polyharmonic , Robin boundary value problems , Schwarz

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium


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Vol.52 • No. 2 • April 2022
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