### 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 |

### Keywords

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

### ASJC Scopus subject areas

- Geometry and Topology

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## Cite this

Hieber, M., Kozono, H., Seyfert, A., Shimizu, S., & Yanagisawa, T. (2019). A Characterization of Harmonic L

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