The L p -Helmholtz projection in finite cylinders

Tobias Nau

Czechoslovak Mathematical Journal (2015)

  • Volume: 65, Issue: 1, page 119-134
  • ISSN: 0011-4642

Abstract

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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.

How to cite

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Nau, Tobias. "The $L^p$-Helmholtz projection in finite cylinders." Czechoslovak Mathematical Journal 65.1 (2015): 119-134. <http://eudml.org/doc/270035>.

@article{Nau2015,
abstract = {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.},
author = {Nau, Tobias},
journal = {Czechoslovak Mathematical Journal},
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; Helmholtz projection; Helmholtz decomposition; weak Neumann problem; periodic boundary conditions; finite cylinder; cylindrical space domain; -space; operator-valued Fourier multiplier; -boundedness; reflection technique; fluid dynamics},
language = {eng},
number = {1},
pages = {119-134},
publisher = {Institute of Mathematics, Academy of Sciences of the Czech Republic},
title = {The $L^p$-Helmholtz projection in finite cylinders},
url = {http://eudml.org/doc/270035},
volume = {65},
year = {2015},
}

TY - JOUR
AU - Nau, Tobias
TI - The $L^p$-Helmholtz projection in finite cylinders
JO - Czechoslovak Mathematical Journal
PY - 2015
PB - Institute of Mathematics, Academy of Sciences of the Czech Republic
VL - 65
IS - 1
SP - 119
EP - 134
AB - 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.
LA - eng
KW - 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; Helmholtz projection; Helmholtz decomposition; weak Neumann problem; periodic boundary conditions; finite cylinder; cylindrical space domain; -space; operator-valued Fourier multiplier; -boundedness; reflection technique; fluid dynamics
UR - http://eudml.org/doc/270035
ER -

References

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