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

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

*この研究の対応する著者

1 被引用数 (Scopus)

## 抄録

Consider the space of harmonic vector fields u in Lr(Ω ) for 1 < r< ∞ for three dimensional exterior domains Ω with smooth boundaries ∂Ω subject to the boundary conditions u· ν= 0 or u× ν= 0 , where ν denotes the unit outward normal on ∂Ω. Denoting these spaces by Xharr(Ω) and Vharr(Ω), it is shown that, in spite of the lack of compactness of Ω , both of these spaces are finite dimensional and that dimVharr(Ω) equals L for 3 / 2 < r< ∞ and L- 1 for 1 < r≤ 3 / 2. Here L is a number representing topologically invariant quantities of ∂Ω and which in the case of bounded domains coincides with the first Betti number. In contrast to the situation of bounded domains, the dimension of Vharr(Ω) in exterior domains is depending on the Lebesgue exponent r. The critical value of this exponent for exterior domains is determined to be 3/2.

本文言語 English 206 Journal of Geometric Analysis 32 7 https://doi.org/10.1007/s12220-022-00938-8 Published - 2022 7月

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