On the Helmholtz decomposition in general unbounded domains

Abstract

It is well known that the Helmholtz decomposition of L q -spaces fails to exist for certain unbounded smooth planar domains unless q = 2, see [2], [9]. As recently shown [6], the Helmholtz projection does exist for general unbounded domains of uniform C2-type in $${\mathbb{R}^{3}}$$ if we replace the space L q , 1 < q < ∞, by L2 ∩ L q for q > 2 and by L q + L2 for 1 < q < 2. In this paper, we generalize this new approach from the three-dimensional case to the n-dimensional case, n ≥ 2. By these means it is possible to define the Stokes operator in arbitrary unbounded domains of uniform C2-type.