On the Helmholtz decomposition in general unbounded domains

Reinhard Farwig, Hideo Kozono, Hermann Sohr

Research output: Contribution to journalArticlepeer-review

46 Citations (Scopus)


It is well known that the Helmholtz decomposition of Lq -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 ℝ3 if we replace the space Lq , 1 < q < ∞, by L2 ∩ Lq for q > 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. By these means it is possible to define the Stokes operator in arbitrary unbounded domains of uniform C2-type.

Original languageEnglish
Pages (from-to)239-248
Number of pages10
JournalArchiv der Mathematik
Issue number3
Publication statusPublished - 2007 Mar


  • General unbounded domains
  • Helmholtz decomposition
  • Sum and intersection spaces


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