Abstract
In this article we prove for 1 < p < ∞ the existence of the L p -Helmholtz projection in finite cylinders Ω. More precisely, Ω is considered to be given as the Cartesian product of a cube and a bounded domain V having C 1-boundary. Adapting an approach of Farwig (2003), operator-valued Fourier series are used to solve a related partial periodic weak Neumann problem. By reflection techniques the weak Neumann problem in Ω is solved, which implies existence and a representation of the L p -Helmholtz projection as a Fourier multiplier operator.
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Nau, T. The L p-Helmholtz projection in finite cylinders. Czech Math J 65, 119–134 (2015). https://doi.org/10.1007/s10587-015-0163-8
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DOI: https://doi.org/10.1007/s10587-015-0163-8
Keywords
- Helmholtz projection
- Helmholtz decomposition
- weak Neumann problem
- periodic boundary conditions
- finite cylinder
- cylindrical space domain
- L p -space
- operatorvalued Fourier multiplier
- R-boundedness
- reflection technique
- fluid dynamics