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

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

### 抄録

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.

元の言語 English Journal of Geometric Analysis https://doi.org/10.1007/s12220-019-00216-0 Published - 2019 1 1

### Fingerprint

Exterior Domain
Vector Field
Harmonic
Unit normal vector
Simple Closed Curve
Compactness
Disjoint
Denote
Boundary conditions

### ASJC Scopus subject areas

• Geometry and Topology

### これを引用

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abstract = "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.",
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T1 - A Characterization of Harmonic Lr -Vector Fields in Two-Dimensional Exterior Domains

AU - Hieber, Matthias Georg

AU - Kozono, Hideo

AU - Seyfert, Anton

AU - Shimizu, Senjo

AU - Yanagisawa, Taku

PY - 2019/1/1

Y1 - 2019/1/1

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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