TY - JOUR

T1 - The Helmholtz–Weyl decomposition of Lr vector fields for two dimensional exterior domains

AU - Hieber, Matthias

AU - Kozono, Hideo

AU - Seyfert, Anton

AU - Shimizu, Senjo

AU - Yanagisawa, Taku

N1 - Funding Information:
The research of the project was partially supported by JSPS Fostering Joint Research Program (B)-18KK0072. The research of H. Kozono was partially supported by JSPS Grant-in-Aid for Scientific Research (S) 16H06339. The research of S. Shimizu was partially supported by JSPS Grant-in-Aid for Scientific Research (B)-16H03945, MEXT.
Publisher Copyright:
© 2020, Mathematica Josephina, Inc.

PY - 2021/5

Y1 - 2021/5

N2 - 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.

AB - 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.

KW - Exterior domains

KW - Harmonic vector fields

KW - Helmholtz–Weyl decomposition

KW - Stream functions and scalar potentials

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U2 - 10.1007/s12220-020-00473-4

DO - 10.1007/s12220-020-00473-4

M3 - Article

AN - SCOPUS:85088311063

SN - 1050-6926

VL - 31

SP - 5146

EP - 5165

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

IS - 5

ER -