$$H^\infty $$-calculus for the Stokes operator with Hodge, Navier, and Robin boundary conditions on unbounded domains
Peer Christian Kunstmann · Integral Equations and Operator Theory · 2025
Abstract We study the Stokes operator with Hodge, Navier, and Robin boundary conditions on domains $$\Omega \subseteq \mathbb {R}^d$$ Ω ⊆ R d that are uniformly $$C^{2,1}$$ C 2 , 1 . Starting with the Hodge Laplacian we etablish a bounded Hörmander functional calculus for the Stokes operator with Hodge boundary conditions. This entails a Hörmander functional calculus and boundedness of the $$H^\infty $$ H ∞ -calculus in spaces of soleniodal vector fields for the Stokes operator with Hodge boundary conditions. We then establish boundedness of the $$H^\infty $$ H ∞ -calculus for Stokes operators with Navier type conditions via Robin type perturbations of Hodge boundary conditions. This implies maximal $$L^p$$ L p -regularity for these operators and results on fractional domain spaces. Our results cover certain non-Helmholtz domains.