Advances in Differential Equations

Weighted $L^q$-Helmholtz decompositions in infinite cylinders and in infinite layers

Reinhard Farwig

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The aim of this paper is to present a new joint approach to the Helmholtz decomposition in infinite cylinders and in infinite layers $\Omega=\Sigma {\times} {\mathbb R}^k$ in the function space $L^q({\mathbb R}^k;L^r({\Sigma}))$ using even arbitrary Muckenhoupt weights with respect to $x'\in \Sigma\subset {\mathbb R}^{n-k}$ and, if possible, exponential weights with respect to $x''\in {\mathbb R}^k, \;1\leq k\leq n-1,\;n\geq 2.$ For $n=2$, we get the Helmholtz decomposition for a strip, for $n=3$ in an infinite cylinder or an infinite layer and for $n>3$ in some (non--physical) unbounded domains of cylinder or layer type. The proof based on a weak Neumann problem uses a partial Fourier transform and operator--valued multiplier functions, the ${\mathcal R}$--boundedness of the family of multiplier operators and an extrapolation property in weighted $L^q$--spaces.

Article information

Adv. Differential Equations, Volume 8, Number 3 (2003), 357-384.

First available in Project Euclid: 19 December 2012

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 35Q30: Navier-Stokes equations [See also 76D05, 76D07, 76N10]
Secondary: 35J20: Variational methods for second-order elliptic equations 35J25: Boundary value problems for second-order elliptic equations 76D05: Navier-Stokes equations [See also 35Q30]


Farwig, Reinhard. Weighted $L^q$-Helmholtz decompositions in infinite cylinders and in infinite layers. Adv. Differential Equations 8 (2003), no. 3, 357--384.

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