Skip to main content
Log in

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

  • Published:
The Journal of Geometric Analysis Aims and scope Submit manuscript


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 \(\partial \Omega \).

This is a preview of subscription content, log in via an institution to check access.

Access this article

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Similar content being viewed by others


  1. Abraham, R., Marsden, J., Ratiu, T.: Manifolds, Tensor Analysis and Applications. Applied Mathematical Sciences. Springer, New York (1988)

    Book  Google Scholar 

  2. Fang, D., Hieber, M., Zi, R.: Global existence results for Oldroyd-B model fluids in exterior domains; the case of non-small computing parameters. Math. Ann. 357, 687–709 (2013)

    Article  MathSciNet  Google Scholar 

  3. Foias, C., Temam, R.: Remarques sur les equations de Navier-Stokes stationaires et les phenomenes successifs de bifurcations. Ann. Scu. Norm. Super. Pisa 5, 29–63 (1978)

    MATH  Google Scholar 

  4. Galdi, G.P.: An Introduction to the Mathematical Theory of the Navier-Stokes Equations. Springer Monographs in Mathematics, vol. 2. Springer, New York (2011)

    Google Scholar 

  5. Günter, N.M.: Potemtial Theory. Frederick Ungar Publishing, New York (1967)

    Google Scholar 

  6. Hieber, M., Kozono, H., Seyfert, A., Shimizu, S., Yanagisawa, T.: A characterization of harmonic \(L^r\)-vector fields in three dimensional exterior domains. Submitted (2018)

  7. Kikuchi, K.: Exterior problem for the two-dimensional Euler equation. J. Fac. Sci. Univ. Tokyo Sect. IA Math. 30(1), 63–92 (1983)

    MathSciNet  MATH  Google Scholar 

  8. Kikuchi, K.: The existence and uniqueness of nonstationary ideal incompressible flow in exterior domains in \(\mathbb{R}^3\). J. Math. Soc. Japan 38(4), 575–598 (1986)

    Article  MathSciNet  Google Scholar 

  9. Korobkov, M., Pileckas, K., Russo, R.: Solution of Leray’s problem for stationary Navier-Stokes equations in plane and axially symmetric domains. Ann. Math. 181, 769–807 (2015)

    Article  MathSciNet  Google Scholar 

  10. Kozono, H., Sohr, H.: On a new class of generalized solutions for the Stokes equations in exterior domains. Ann Scu. Norm. Super. Pisa 19, 155–181 (1992)

    MathSciNet  MATH  Google Scholar 

  11. Kozono, H., Yanagisawa, T.: \(L^r\)-variational inequality for vector fields and the Helmholtz-Weyl decomposition in bounded domains. Indiana Univ. Math. J. 58, 1853–1920 (2009)

    Article  MathSciNet  Google Scholar 

  12. Kozono, H., Yanagisawa, T.: Leray’s problem on the stationary Navier–Stokes equations with inhomogeneous boundary data. Math. Z. 262, 27–39 (2009)

    Article  MathSciNet  Google Scholar 

  13. Kozono, H., Yanagisawa, T.: Nonhomogeneous boundary value problems for stationary Navier–Stokes equations in a multiply connected bounded domain. Pac. J. Math. 243, 127–150 (2009)

    Article  MathSciNet  Google Scholar 

  14. Morrey, C.B.: Multiple Integrals in the Calculus of Variations Grudlehren der mathematische Wissenschaften 130. Springer, Heidelberg (1966)

    Book  Google Scholar 

  15. Simader, C.G., Sohr, H.: A new approach to the Helmholtz decomposition and the Neumann problem in Lq-spaces for bounded and exterior domains. In: Galdi, G.P. (ed.) Mathematical Problems relating to the Navier–Stokes Equations, Advances in Mathematics for Applied Sciences, pp. 1–35. World Scientific, Singapore (1992)

    MATH  Google Scholar 

  16. Simader, C.G., Sohr, H.: The Dirichlet Problem for the Laplacian in Bounded and Unbounded Domains Pitman Research Notes in Mathematics, vol. 360. Longman, Harlow (1996)

    MATH  Google Scholar 

Download references


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.

Author information

Authors and Affiliations


Corresponding author

Correspondence to Senjo Shimizu.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

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).

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI:


Mathematics Subject Classification