A Characterization of Harmonic Lr -Vector Fields in Two-Dimensional Exterior Domains

Matthias Hieber, Hideo Kozono, Anton Seyfert, Senjo Shimizu*, Taku Yanagisawa

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

5 Citations (Scopus)


Consider the space of harmonic vector fields h in Lr(Ω) for 1 < r< ∞ in the two-dimensional exterior domain Ω with the smooth boundary ∂Ω subject to the boundary conditions h· ν= 0 or h∧ ν= 0 , where ν denotes the unit outward normal to ∂Ω. Denoting these spaces by Xharr(Ω) and Vharr(Ω), respectively, it is shown that, in spite of the lack of compactness of Ω , both of these spaces are finite dimensional and that their dimension of both spaces coincides with L for 2 < r< ∞ and L- 1 for 1 < r≤ 2. Here L is the number of disjoint simple closed curves consisting of the boundary ∂Ω.

Original languageEnglish
Pages (from-to)3742-3759
Number of pages18
JournalJournal of Geometric Analysis
Issue number4
Publication statusPublished - 2020 Dec 1


  • Betti number
  • Exterior domains
  • Harmonic vector fields
  • Helmholtz–Weyl decomposition

ASJC Scopus subject areas

  • Geometry and Topology


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