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January, 2015 Weak Neumann implies $H^\infty$ for Stokes
Matthias GEIßERT, Peer Christian KUNSTMANN
J. Math. Soc. Japan 67(1): 183-193 (January, 2015). DOI: 10.2969/jmsj/06710183


Let $\Omega\subset {\mathbb R}^n$ be a domain with uniform $C^3$ boundary and assume that the Helmholtz decomposition exists in ${\mathbb L}^q(\Omega):=L^q(\Omega)^n$ for some $q\in(1,\infty)$. We show that a suitable translate of the Stokes operator admits a bounded ${\cal H}^\infty$-calculus in ${\mathbb L}_\sigma^p(\Omega)$ for $p\in(\min\{q,q'\},\max\{q,q'\})$. For the proof we use a recent maximal regularity result for the Stokes operator on such domains ([GHHS12]) and an abstract result for the ${\cal H}^\infty$-calculus in complemented subspaces ([KKW06], [KW13]).


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Matthias GEIßERT. Peer Christian KUNSTMANN. "Weak Neumann implies $H^\infty$ for Stokes." J. Math. Soc. Japan 67 (1) 183 - 193, January, 2015.


Published: January, 2015
First available in Project Euclid: 22 January 2015

zbMATH: 1317.35173
MathSciNet: MR3304018
Digital Object Identifier: 10.2969/jmsj/06710183

Primary: 35Q30 , 47A60

Keywords: $H^\infty$-functional calculus , fractional powers , general unbounded domains , Helmholtz decomposition , Stokes operator

Rights: Copyright © 2015 Mathematical Society of Japan

Vol.67 • No. 1 • January, 2015
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