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Annals of Global Analysis and Geometry

, Volume 35, Issue 1, pp 83–90 | Cite as

A Trudinger type inequality for maps into a Riemannian manifold

  • Roger MoserEmail author
Original Paper
  • 91 Downloads

Abstract

The tension field of a map into a Riemannian manifold is the equivalent to the Laplacian of a function. However in contrast to the latter, the tension field is given by a nonlinear differential operator. Nevertheless, it permits an extension of a well-known Trudinger inequality that involves an Orlicz space for a function with exponential growth.

Keywords

Trudinger inequality Tension field 

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References

  1. 1.
    Adams D.R.: A sharp inequality of J. Moser for higher order derivatives. Ann. Math. 128(2), 385–398 (1988)CrossRefGoogle Scholar
  2. 2.
    Aubin T.: Some nonlinear problems in Riemannian geometry. Springer Monographs in Mathematics, Springer-Verlag, Berlin (1998)zbMATHGoogle Scholar
  3. 3.
    Chang S.-Y.A.: Conformal invariants and partial differential equations. Bull. Amer. Math. Soc. (N.S.) 42, 365–393 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  4. 4.
    Chang S.-Y.A., Yang P.C.: The inequality of Moser and Trudinger and applications to conformal geometry. Comm. Pure Appl. Math. 56, 1135–1150 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    John F., Nirenberg L.: On functions of bounded mean oscillation. Comm. Pure Appl. Math. 14, 415–426 (1961)zbMATHCrossRefMathSciNetGoogle Scholar
  6. 6.
    Jost J.: Riemannian Geometry and Geometric Analysis, 4th edn. Universitext Springer-Verlag, Berlin (2005)Google Scholar
  7. 7.
    Montaldo S., Oniciuc C.: A short survey on biharmonic maps between Riemannian manifolds. Rev. Un. Mat. Argentina 47(2), 1–22 (2006)zbMATHMathSciNetGoogle Scholar
  8. 8.
    Moser J.: A sharp form of an inequality by N. Trudinger. Indiana Univ. Math. J. 20, 1077–1092 (1970)CrossRefMathSciNetGoogle Scholar
  9. 9.
    Moser R.: Regularity for the approximated harmonic map equation and application to the heat flow for harmonic maps. Math. Z. 243, 263–289 (2003)zbMATHCrossRefMathSciNetGoogle Scholar
  10. 10.
    Moser R.: Energy concentration for almost harmonic maps and the Willmore functional. Math. Z. 251, 293–311 (2005)zbMATHCrossRefMathSciNetGoogle Scholar
  11. 11.
    Moser R.: Remarks on the regularity of biharmonic maps in four dimensions. Comm. Pure Appl. Math. 59, 317–329 (2006)zbMATHCrossRefMathSciNetGoogle Scholar
  12. 12.
    Moser, R.: A variational problem pertaining to biharmonic maps. Preprint (2008)Google Scholar
  13. 13.
    Price P.: A monotonicity formula for Yang-Mills fields. Manuscripta Math. 43, 131–166 (1983)zbMATHCrossRefMathSciNetGoogle Scholar
  14. 14.
    Trudinger N.S.: On imbeddings into Orlicz spaces and some applications. J. Math. Mech. 17, 473–483 (1967)zbMATHMathSciNetGoogle Scholar

Copyright information

© Springer Science+Business Media B.V. 2008

Authors and Affiliations

  1. 1.Department of Mathematical SciencesUniversity of BathBathUK

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