The Helmholtz–Weyl decomposition of Lr vector fields for two dimensional exterior domains

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

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


Let Ω be a two-dimensional exterior domain with smooth boundary ∂Ω and 1 < r< ∞. Then Lr(Ω) 2 allows a Helmholtz–Weyl decomposition, i.e., for every u∈ Lr(Ω) 2 there exist h∈Xharr(Ω), w∈ H˙ 1,r(Ω) and p∈ H˙ 1,r(Ω) such that u=h+rotw+∇p.The function h can be chosen alternatively also from Vharr(Ω), another space of harmonic vector fields subject to different boundary conditions. These spaces Xharr(Ω) and Vharr(Ω) of harmonic vector fields are known to be finite dimensional. The above decomposition is unique if and only if 1 < r≦ 2 , while in the case 2 < r< ∞, uniqueness holds only modulo a one dimensional subspace of Lr(Ω) 2. The corresponding result for the three dimensional setting was proved in our previous paper, where in contrast to the two dimensional case, there are two threshold exponents, namely r= 3 / 2 and r= 3. In our two dimensional situation, r= 2 is the only critical exponent, which determines the validity of a unique Helmholtz–Weyl decomposition.

Original languageEnglish
Pages (from-to)5146-5165
Number of pages20
JournalJournal of Geometric Analysis
Issue number5
Publication statusPublished - 2021 May


  • Exterior domains
  • Harmonic vector fields
  • Helmholtz–Weyl decomposition
  • Stream functions and scalar potentials


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