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.