### Abstract

Consider the space of harmonic vector fields h in L^{r}(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.

Original language | English |
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Journal | Journal of Geometric Analysis |

DOIs | |

Publication status | Published - 2019 Jan 1 |

### Fingerprint

### Keywords

- Betti number
- Exterior domains
- Harmonic vector fields
- Helmholtz–Weyl decomposition

### ASJC Scopus subject areas

- Geometry and Topology

### Cite this

^{r}-Vector Fields in Two-Dimensional Exterior Domains.

*Journal of Geometric Analysis*. https://doi.org/10.1007/s12220-019-00216-0

**A Characterization of Harmonic L ^{r} -Vector Fields in Two-Dimensional Exterior Domains.** / Hieber, Matthias Georg; Kozono, Hideo; Seyfert, Anton; Shimizu, Senjo; Yanagisawa, Taku.

Research output: Contribution to journal › Article

^{r}-Vector Fields in Two-Dimensional Exterior Domains',

*Journal of Geometric Analysis*. https://doi.org/10.1007/s12220-019-00216-0

^{r}-Vector Fields in Two-Dimensional Exterior Domains. Journal of Geometric Analysis. 2019 Jan 1. https://doi.org/10.1007/s12220-019-00216-0

}

TY - JOUR

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

UR - http://www.scopus.com/inward/record.url?scp=85067230565&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85067230565&partnerID=8YFLogxK

U2 - 10.1007/s12220-019-00216-0

DO - 10.1007/s12220-019-00216-0

M3 - Article

AN - SCOPUS:85067230565

JO - Journal of Geometric Analysis

JF - Journal of Geometric Analysis

SN - 1050-6926

ER -