The Journal of Geometric Analysis

, Volume 6, Issue 1, pp 91–112

Regularity forn-harmonic maps

  • Libin Mou
  • Paul Yang
Article

Abstract

Here we obtain everywhere regularity of weak solutions of some nonlinear elliptic systems with borderline growth, includingn-harmonic maps between manifolds or map with constant volumes. Other results in this paper include regularity up to the boundary and a removability theorem for isolated singularities.

Math Subject Classification

58E20 35J60 35J70 42B30 

Key Words and Phrases

n-harmonic maps borderline growth removable singularities interior and boundary regularity volume constraint compensation H1 theory 

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References

  1. [AR]
    Adams, R. A.Sobolev Spaces, Academic Press, New York, 1975.MATHGoogle Scholar
  2. [B]
    Bethuel, F. On the singular set of stationary harmonic maps.Manuscripta Math. 28, 417–443 (1993).CrossRefMathSciNetGoogle Scholar
  3. [CLMS1]
    Coiffman, R., Lions, P.-L., Meyer, Y., and Semmes, S. Compacité par compensation et espaces de Hardy.C. R. Acad. Sci. Paris 311, 519–524 (1989).Google Scholar
  4. [CLMS2]
    Coiffman, R., Lions, P.-L., Meyer, Y., and Semmes, S. Compensated compactness and Hardy space.J. Math. Pures Appl. 72, 247–286 (1993).MathSciNetGoogle Scholar
  5. [CL]
    Costa, D., and Liao, G. On the removability of a singular submanifold for weakly harmonic maps.J. Fac. Sci. Univ. Tokyo, Sec. 1A 35(2), 321–344 (1988).MathSciNetMATHGoogle Scholar
  6. [DF]
    Duzaar, F., and Fuchs, M. On removable singularities of p-harmonic maps.Ann. Inst. Henri Poincaré 7(5), 385–405 (1990).MathSciNetMATHGoogle Scholar
  7. [EL]
    Evans, L. C. Partial regularity for stationary harmonic maps into spheres.Arch. Rat. Mech. Anal. 116, 101–113 (1991).CrossRefMATHGoogle Scholar
  8. [FS]
    Fefferman, C., and Stein, E.H p spaces of several variables.Acta Math. 129, 137–193 (1973).CrossRefMathSciNetGoogle Scholar
  9. [G]
    Giaquinta, M.Multiple Integrals in the Calculus of Variations and Nonlinear Elliptic Systems, Princeton University Press, Princeton, NJ, 1989.Google Scholar
  10. [Gr]
    Grüter, M. Regularity of weak H-surfaces.J. Reine Angew. Math. 329, 1–15(1981).MathSciNetMATHGoogle Scholar
  11. [HL1]
    Hardt, R., and Lin, F. H. Mappings minimizing theL p norm of the gradient.Comm. Pure Appl. Math. 40, 555–588 (1987).CrossRefMathSciNetMATHGoogle Scholar
  12. [HL2]
    Hardt, R., and Lin, F. H. Personal communication (1992).Google Scholar
  13. [HF1]
    Hélein, F. Regularite des applications faiblement harmoniques entre une surface et variete riemannienne.CRAS, Paris 312, 591–596(1991).MATHGoogle Scholar
  14. [HF2]
    Hélein, F. Regularity of weakly harmonie maps from a surface in a manifold with symmetries.Manuscripta Math. 70, 203–218 (1991).CrossRefMathSciNetMATHGoogle Scholar
  15. [JN]
    John, F., and Nirenberg, L. On functions of bounded mean oscillations.Comm. Pure Appl. Math. 14, 415–426 (1964).CrossRefMathSciNetGoogle Scholar
  16. [L]
    Lewis, J. Smoothness of certain degenerate elliptic systems.Proc. Amer. Math. Soc. 80, 259–265 (1980).CrossRefMathSciNetMATHGoogle Scholar
  17. [LG1]
    Liao, G. Regularity theorem for harmonic maps with small energy.J. Differential Geom. 22, 233–241 (1985).MathSciNetMATHGoogle Scholar
  18. [LG2]
    Liao, G. A study of regularity problem of harmonic maps.Pacific J. Math. 130 (1987).Google Scholar
  19. [LS]
    Luckhaus, S. C.1,ε-regularity for energy minimizing Hölder continuous p-harmonic maps between Riemannian manifolds.Indiana Univ. Math. J. 37, 349–367 (1989).CrossRefMathSciNetGoogle Scholar
  20. [MC]
    Morrey, C.Multiple Integrals in the Calculus of Variations, Springer-Verlag, Heidelberg, 1966.MATHGoogle Scholar
  21. [ML]
    Mou, L. Removability of singular sets of harmonic maps.Arch. Rat. Mech. Anal. 127(3), 199–217 (1994).CrossRefMathSciNetMATHGoogle Scholar
  22. [MY1]
    Mou, L., and Yang, P. Existence ofn-harmonic maps with prescribed volumes. Preprint.Google Scholar
  23. [QJ]
    Qing, J. Boundary regularity of harmonic maps from surfaces.J. Funct. Anal. 114, 458–466 (1993).CrossRefMathSciNetMATHGoogle Scholar
  24. [RT]
    Riviere, T. Everywhere discontinuous harmonic maps into spheres.Acta Math. 175(2), 197–226 (1995).CrossRefMathSciNetMATHGoogle Scholar
  25. [SaU]
    Sacks, J., and Uhlenbeck, K. The existence of minimal immersions of 2-spheres.Ann. of Math. (2)113, 1–24 (1981).CrossRefMathSciNetGoogle Scholar
  26. [SR]
    Schoen, R. Analytic aspects of the harmonic map problem. InSeminar on Nonlinear P.D.E., S. S. Chern, ed., Springer-Verlag, New York, Berlin, 1984.Google Scholar
  27. [SU1]
    Schoen, R., and Uhlenbeck, K. A regularity theory for harmonic maps.J. Differential Geom. 17, 307–335 (1982).MathSciNetGoogle Scholar
  28. [SU2]
    Schoen, R., and Uhlenbeck, K. Boundary regularity and the Dirichlet problem for harmonic maps.J. Differential Geom. 18, 253–268 (1983).MathSciNetMATHGoogle Scholar
  29. [Ss]
    Semmes, S. A primer on Hardy spaces, and some remarks on a theorem of Evans and Müller.Comm. P.D.E. 19, 277–319(1994).CrossRefMathSciNetMATHGoogle Scholar
  30. [SL]
    Simon, L. The singular set of minimal submanifolds and harmonic maps. Preprint (1992).Google Scholar
  31. [T]
    Tolksdorff, P. Regularity for a more general class of quasi-linear elliptic equations.J. Differential Eq. 51, 126–150 (1984).CrossRefGoogle Scholar
  32. [TW]
    Toro, T., and Wang, C. Y. Compactness properties of weaklyp-harmonic mapping into homogeneous spaces.Indiana Univ. Math. J. 44(1) (1995).Google Scholar
  33. [UK]
    Uhlenbeck, K. Regularity of a class of nonlinear elliptic systems.Acta Math. 138, 219–240 (1970).CrossRefMathSciNetGoogle Scholar
  34. [WH]
    Wente, H. An existence theorem for surfaces of constant mean curvature.J. Math. Anal. Appl. 26, 318–344 (1969).CrossRefMathSciNetMATHGoogle Scholar

Copyright information

© Mathematica Josephina, Inc. 1996

Authors and Affiliations

  • Libin Mou
    • 1
  • Paul Yang
    • 1
  1. 1.Department of MathematicsUniversity of Southern CaliforniaLos Angeles
  2. 2.Department of MathematicsBradley UniversityPeoria

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