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

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

*Corresponding author for this work

Research output: Contribution to journalArticlepeer-review

Abstract

Consider the space of harmonic vector fields u in Lr(Ω ) for 1 < r< ∞ for three dimensional exterior domains Ω with smooth boundaries ∂Ω subject to the boundary conditions u· ν= 0 or u× ν= 0 , where ν denotes the unit outward normal on ∂Ω. Denoting these spaces by Xharr(Ω) and Vharr(Ω), it is shown that, in spite of the lack of compactness of Ω , both of these spaces are finite dimensional and that dimVharr(Ω) equals L for 3 / 2 < r< ∞ and L- 1 for 1 < r≤ 3 / 2. Here L is a number representing topologically invariant quantities of ∂Ω and which in the case of bounded domains coincides with the first Betti number. In contrast to the situation of bounded domains, the dimension of Vharr(Ω) in exterior domains is depending on the Lebesgue exponent r. The critical value of this exponent for exterior domains is determined to be 3/2.

Original languageEnglish
Article number206
JournalJournal of Geometric Analysis
Volume32
Issue number7
DOIs
Publication statusPublished - 2022 Jul

Keywords

  • Betti numbers
  • Exterior domains
  • Harmonic vector fields
  • Helmholtz–Weyl decomposition
  • Jump condition

ASJC Scopus subject areas

  • Geometry and Topology

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