# Article

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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
Summary:
In this article we prove for $1<p<\infty$ 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.
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