Lr-Helmholtz-Weyl decomposition for three dimensional exterior domains

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

Research output: Contribution to journalArticlepeer-review

4 Citations (Scopus)


In this article the Helmholtz-Weyl decomposition in three dimensional exterior domains is established within the Lr-setting for 1<r<∞. In fact, given an Lr-vector field u, there exist h∈Xharr(Ω), w∈H˙1,r(Ω)3 with divw=0 and p∈H˙1,r(Ω) such that u may be decomposed uniquely as u=h+rotw+∇p. If for the given Lr-vector field u, its harmonic part h is chosen from Vharr(Ω), then a decomposition similar to the above one is established, too. However, its uniqueness holds in this case only for the case 1<r<3. The proof given relies on an Lr-variational inequality allowing to construct w∈H˙1,r(Ω)3 and p∈H˙1,r(Ω) for given u∈Lr(Ω)3 as weak solutions to certain elliptic boundary value problems.

Original languageEnglish
Article number109144
JournalJournal of Functional Analysis
Issue number8
Publication statusPublished - 2021 Oct 15


  • Exterior domains
  • Harmonic vector fields
  • Helmholtz-Weyl decomposition
  • Vector and scalar potentials


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