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A Characterization of Harmonic \(L^r\)-Vector Fields in Two-Dimensional Exterior Domains

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Abstract

Consider the space of harmonic vector fields h in \(L^r(\Omega )\) for \(1<r<\infty \) in the two-dimensional exterior domain \(\Omega \) with the smooth boundary \(\partial \Omega \) subject to the boundary conditions \(h\cdot \nu =0\) or \(h\wedge \nu =0\), where \(\nu \) denotes the unit outward normal to \(\partial \Omega \). Denoting these spaces by \(X^r_{\tiny {\text{ har }}}(\Omega )\) and \(V^r_{\tiny {\text{ har }}}(\Omega )\), respectively, it is shown that, in spite of the lack of compactness of \(\Omega \), both of these spaces are finite dimensional and that their dimension of both spaces coincides with L for \(2< r<\infty \) and \(L-1\) for \(1<r\le 2\). . Here L is the number of disjoint simple closed curves consisting of the boundary \(\partial \Omega \).

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Acknowledgements

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.

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Correspondence to Senjo Shimizu.

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Hieber, M., Kozono, H., Seyfert, A. et al. A Characterization of Harmonic \(L^r\)-Vector Fields in Two-Dimensional Exterior Domains. J Geom Anal 30, 3742–3759 (2020). https://doi.org/10.1007/s12220-019-00216-0

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  • DOI: https://doi.org/10.1007/s12220-019-00216-0

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