Skip to main content
SpringerLink
Log in

A Reformulation of the Biharmonic Map Equation

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

Abstract

The known Euler–Lagrange equation for (intrinsic) biharmonic maps is unsuitable for the study of some of the critical points of the corresponding functional, as it requires too much regularity. We derive and discuss a variant of the equation that does not have this shortcoming.

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

Access this article

Log in via an institution

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

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Hélein, F.: Regularity of weakly harmonic maps from a surface into a manifold with symmetries. Manuscr. Math. 70, 203–218 (1991)

    Article  MATH  Google Scholar 

  2. Hornung, P., Moser, R.: Intrinsically biharmonic maps into homogeneous spaces. Adv. Calc. Var. 5, 411–425 (2012)

    Article  MATH  MathSciNet  Google Scholar 

  3. Jäger, W., Kaul, H.: Uniqueness and stability of harmonic maps and their Jacobi fields. Manuscr. Math. 28, 269–291 (1979)

    Article  MATH  Google Scholar 

  4. Jiang, G.Y.: 2-Harmonic maps and their first and second variational formulas. Chin. Ann. Math., Ser. A 7, 389–402 (1986)

    MATH  Google Scholar 

  5. Jost, J.: Riemannian Geometry and Geometric Analysis. Universitext. Springer, Berlin (1995)

    Book  MATH  Google Scholar 

  6. Moser, R.: A variational problem pertaining to biharmonic maps. Commun. Partial Differ. Equ. 33, 1654–1689 (2008)

    Article  MATH  Google Scholar 

  7. Nirenberg, L.: An extended interpolation inequality. Ann. Sc. Norm. Super. Pisa, Cl. Sci. 20, 733–737 (1966)

    MATH  MathSciNet  Google Scholar 

  8. Rivière, T.: Everywhere discontinuous harmonic maps into spheres. Acta Math. 175, 197–226 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Schoen, R., Uhlenbeck, K.: A regularity theory for harmonic maps. J. Differ. Geom. 17, 307–335 (1982)

    MATH  MathSciNet  Google Scholar 

Download references

Acknowledgements

P. Hornung is supported by the DFG through grant no. HO-4697/1-1. R. Moser is partially supported by the EPSRC through grant EP/F048769/1.

Author information

Authors and Affiliations

  1. Max Planck Institute for Mathematics in the Sciences, Inselstraße 22, 04103, Leipzig, Germany

    Peter Hornung

  2. Department of Mathematical Sciences, University of Bath, Bath, BA2 7AY, UK

    Roger Moser

Authors
  1. Peter Hornung
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Roger Moser
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Roger Moser.

Additional information

Communicated by Jiaping Wang.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Hornung, P., Moser, R. A Reformulation of the Biharmonic Map Equation. J Geom Anal 24, 1201–1210 (2014). https://doi.org/10.1007/s12220-012-9369-2

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12220-012-9369-2

Keywords

Mathematics Subject Classification

Search

Navigation