Weak Neumann implies H^∞ for Stokes

Abstract

Let Ω ⊂ ℝⁿ be a domain with uniform C³ boundary and assume that the Helmholtz decomposition exists in $\mathbb L$^q(Ω) := L^q(Ω)ⁿ for some q ∈ (1,∞). We show that a suitable translate of the Stokes operator admits a bounded{\cal H} ^∞-calculus in $\mathbb L$_σ^p(Ω) for p ∈ (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}^∞-calculus in complemented subspaces ([KKW06], [KW13]).