The Helmholtz decomposition in arbitrary unbounded domains : a theory beyond L2
Reinhard Farwig, Hideo Kozono, Hermann Sohr · 2005
Abstract. It is well known that the usual Lq-theory of the Stokes operator valid for bounded or exterior domains cannot be extended to arbitrary unbounded domains Ω ⊂ Rn when q 6 = 2. One reason is given by the Helmholtz projection which fails to exist for certain unbounded smooth planar domains unless q = 2. However, as recently shown [6], the Helmholtz projection does exist for general unbounded domains in R3 if we replace the space Lq, 1 2 and by Lq +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. Key words. Helmholtz decomposition, Helmholtz projection, general unbounded domains, domains of uniform C1-type, intersection spaces, sum spaces AMS subject classifications. 35Q30, 76D05