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

Abstract

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
JournalJournal of Geometric Analysis
DOIs
Publication statusAccepted/In press - 2020

Keywords

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

ASJC Scopus subject areas

  • Geometry and Topology

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