Remarks on weakly stationary maps into spheres characterized by wedge product

  • Masashi Misawa
  • Nobumitsu NakauchiEmail author


In Chen (Math Z 201:69–74, 1989, Math Z 215:25–35, 1994), Chen first used the wedge product for weak solutions of the harmonic map flow and the p-harmonic one into the spheres. The notion of the wedge product is useful in studying such weak solutions into the spheres. In this paper we consider weak solutions of systems of more general differential equations. Furthermore we clarify an essential structure of general systems of equations necessary for an application of wedge product.


Elliptic system Wedge product Weak solution Sphere Harmonic map Symphonic map 

Mathematics Subject Classification

Primary Secondary: 58E99 58E20 53C43 



This work was partially supported by the Grant-in-Aid for Scientific Research (C) No.15K04962 (Misawa) and No.15K04846 (Nakauchi) at Japan Society for the Promotion of Science.


  1. 1.
    Chen, Y.: The weak solutions to the evolution problems of harmonic maps. Math. Z. 201, 69–74 (1989)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Chen, Y., Hong, M.-C., Hungerbühler, N.: Heat flow of \(p\)-harmonic maps with values into the spheres. Math. Z. 215, 25–35 (1994)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Eells, J., Lemaire, L.: A report on harmonic maps. Bull. Lond. Math. Soc. 10, 1–68 (1978)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Eells, J., Lemaire, L.: Another report on harmonic maps. Bull. Lond. Math. Soc. 20, 385–524 (1988)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Giaquinta, M.: Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems. Princeton University Press, Princeton (1983)zbMATHGoogle Scholar
  6. 6.
    Kawai, S., Nakauchi, N.: Some results for stationary maps of a functional related to pullback metrics. Nonlinear Anal. 74, 2284–2295 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  7. 7.
    Kawai, S., Nakauchi, N.: Stability of stationary maps of a functional related to pullbacks of metrics. Differ. Geom. Appl. 44, 161–177 (2016)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Kawai, S., Nakauchi, N.: Weak conformality of stable stationary maps for a functional related to conformality. Differ. Geom. Appl. 31, 151–165 (2013)MathSciNetCrossRefzbMATHGoogle Scholar
  9. 9.
    Misawa, M., Nakauchi, N.: A Hölder continuity of minimizing symphonic maps. Nonlinear Anal. 75, 5971–5974 (2012)MathSciNetCrossRefzbMATHGoogle Scholar
  10. 10.
    Misawa, M., Nakauchi, N.: A Hölder continuity of symphonic maps into the spheres. Calc. Var. Partial Differ. Equ. 55, 1–20 (2016)CrossRefzbMATHGoogle Scholar
  11. 11.
    Misawa, M., Nakauchi, N.: Global existence for the heat flow of symphonic maps into spheres, to appear in Advances in Differential EquationsGoogle Scholar
  12. 12.
    Misawa, M., Nakauchi, N.: in preparationGoogle Scholar
  13. 13.
    Nakauchi, N., Takakuwa, S.: Symphonic join of maps between the spheres. Nonlinear Anal. 108, 87–98 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  14. 14.
    Nakauchi, N., Takenaka, Y.: A variational problem for pullback metrics. Ricerche di Matematica 60, 219–235 (2011)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer-Verlag Italia S.r.l., part of Springer Nature 2018

Authors and Affiliations

  1. 1.Department of Mathematics, Graduate School of Science and TechnologyKumamoto UniversityKumamotoJapan
  2. 2.Graduate School of Sciences and Technology for InnovationYamaguchi UniversityYamaguchiJapan

Personalised recommendations