On the Stokes operator in general unbounded domains

It is known that the Stokes operator is not well-defined in L^q-spaces for certain unbounded smooth domains unless q = 2. In this paper, we generalize a new approach to the Stokes resolvent problem and to maximal regularity in general un-bounded smooth domains from the three-dimensional case, see [7], to the n-dimensional one, n ≥ 2, replacing the space L^q, 1 < q < ∞, by L̃^q where L̃^q = L̃^q ∩ L² for q ≥ 2 and L̃^q = L^q + L² for 1 < q < 2. In particular, we show that the Stokes operator is well-defined in L^q for every unbounded domain of uniform C^1,1-type in Rⁿ, n ≥ 2, satisfies the classical resolvent estimate, generates an analytic semigroup and has maximal regularity.