The Stokes resolvent problem in general unbounded domains
Reinhard Farwig, Hideo Kozono, Hermann Sohr · TUbilio (Technical University of Darmstadt) · 2007
It is well-known that the Helmholtz decomposition of Lq-spaces fails to exist for certain unbounded smooth domains unless q = 2. Hence also the Stokes operator is not well-defined for these domains when q 6 = 2. In this paper, we generalize a new approach to the Stokes problem in general unbounded smooth domains from the three-dimensional case, see [5], to the n-dimensional one, n ≥ 2, replacing the space Lq, 1 < q < ∞, by L̃q where L̃q = Lq ∩ L2 for q ≥ 2 and L̃q = Lq + L2 for 1 < q < 2. In particular, we show that the Stokes operator is well-defined in L̃q for every unbounded domain of uniform C1,1-type in Rn, n ≥ 2, and generates an analytic semigroup.