Journal
ANALYSIS AND MATHEMATICAL PHYSICS
Volume 11, Issue 3, Pages -Publisher
SPRINGER BASEL AG
DOI: 10.1007/s13324-021-00569-2
Keywords
Poly-analytic; Cauchy-Schwarz-Pompeiu representation; Green function; Schwarz; Dirichlet; Neumann boundary value problems; Admissible domain; Ring domain; Bi- and tri-analytic Pompeiu integral operators
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Funding
- Projekt DEAL
- Ministry of Education and Science of the Republic of Kazakhstan [AP08052345]
- German Academic Exchange Service (DAAD) [57440919]
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The paper investigates the boundary value problems for the inhomogeneous polyanalytic equation in planar domains, including the well-posed Schwarz problem, Dirichlet problem, and Neumann problem. While a modification is mentioned for the Schwarz problem, the Dirichlet problem is completely discussed for arbitrary order, and the Neumann problem is only handled for order up to three, with potential generalization to arbitrary order in the future.
There are three basic boundary value problems for the inhomogeneous polyanalytic equation in planar domains, the well-posed iterated Schwarz problem, and further two over-determined iterated problems of Dirichlet and Neumann type. These problems are investigated in planar domains having a harmonic Green function. For the Schwarz problem, treated earlier [u. Aksoy, H. Begehr, A.O. celebi, AV Bitsadze's observation on bianalytic functions and the Schwarz problem. Complex Var Elliptic Equ 64(8): 1257-1274 (2019)], just a modification is mentioned. While the Dirichlet problem is completely discussed for arbitrary order, the Neumann problem is just handled for order up to three. But a generalization to arbitrary order is likely.
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