TY - JOUR

T1 - A Characterization of Harmonic Lr -Vector Fields in Two-Dimensional Exterior Domains

AU - Hieber, Matthias

AU - Kozono, Hideo

AU - Seyfert, Anton

AU - Shimizu, Senjo

AU - Yanagisawa, Taku

N1 - Funding Information:
The work is partially supported by JSPS Fostering Joint Research Program (B) Grant No. 18KK0072. The work of the second author is partially supported by JSPS Grant-in-aid for Scientific Research S #16H06339. The work of the fourth author is partially supported by JSPS Grant-in-aid for Scientific Research B #16H03945.

PY - 2020/12/1

Y1 - 2020/12/1

N2 - Consider the space of harmonic vector fields h in Lr(Ω) for 1 < r< ∞ in the two-dimensional exterior domain Ω with the smooth boundary ∂Ω subject to the boundary conditions h· ν= 0 or h∧ ν= 0 , where ν denotes the unit outward normal to ∂Ω. Denoting these spaces by Xharr(Ω) and Vharr(Ω), respectively, it is shown that, in spite of the lack of compactness of Ω , both of these spaces are finite dimensional and that their dimension of both spaces coincides with L for 2 < r< ∞ and L- 1 for 1 < r≤ 2. Here L is the number of disjoint simple closed curves consisting of the boundary ∂Ω.

AB - Consider the space of harmonic vector fields h in Lr(Ω) for 1 < r< ∞ in the two-dimensional exterior domain Ω with the smooth boundary ∂Ω subject to the boundary conditions h· ν= 0 or h∧ ν= 0 , where ν denotes the unit outward normal to ∂Ω. Denoting these spaces by Xharr(Ω) and Vharr(Ω), respectively, it is shown that, in spite of the lack of compactness of Ω , both of these spaces are finite dimensional and that their dimension of both spaces coincides with L for 2 < r< ∞ and L- 1 for 1 < r≤ 2. Here L is the number of disjoint simple closed curves consisting of the boundary ∂Ω.

KW - Betti number

KW - Exterior domains

KW - Harmonic vector fields

KW - Helmholtz–Weyl decomposition

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U2 - 10.1007/s12220-019-00216-0

DO - 10.1007/s12220-019-00216-0

M3 - Article

AN - SCOPUS:85067230565

VL - 30

SP - 3742

EP - 3759

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

IS - 4

ER -