Geodesic
The $L^p$-Helmholtz projection in finite cylinders
Czechoslovak Mathematical Journal, Tome 65 (2015) no. 1, pp. 119-134.

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In this article we prove for $1$ the existence of the $L^p$-Helmholtz projection in finite cylinders $\Omega $. More precisely, $\Omega $ 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 $\Omega $ is solved, which implies existence and a representation of the $L^p$-Helmholtz projection as a Fourier multiplier operator.
DOI : 10.1007/s10587-015-0163-8
Classification : 35J20, 35J25, 35Q30, 42B15, 46E40
Keywords: Helmholtz projection; Helmholtz decomposition; weak Neumann problem; periodic boundary conditions; finite cylinder; cylindrical space domain; $L^p$-space; operator-valued Fourier multiplier; $\mathcal R$-boundedness; reflection technique; fluid dynamics 
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Nau, Tobias. The $L^p$-Helmholtz projection in finite cylinders. Czechoslovak Mathematical Journal, Tome 65 (2015) no. 1, pp. 119-134. doi : 10.1007/s10587-015-0163-8. http://geodesic.mathdoc.fr/articles/10.1007/s10587-015-0163-8/

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