Approximation in Sobolev Spaces between two manifolds and homotopy groups

  • Fabrice Bethuel
Part of the Progress in Nonlinear Differential Equations and Their Applications book series (PNLDE, volume 4)


We consider two compact manifolds M n and N k and the Sobolev spaces W 1,p (M n , N k ),for 1 < p < n = dim M n . We give a necessary and sufficient condition for smooth maps between M n and N k to be dense in W 1,p (M n , N k ). This condition can be simply stated in terms of homotopy groups, and is π[p](N k )= 0. In cases where such a condition does not hold, we show that we can approximate maps in W 1,p (M n , N k ) by maps smooth except on a singular set which has a simple shape. We consider also the problem of the weak density of smooth maps.


Finite Number Sobolev Space Point Singularity Weak Topology Homotopy Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [Be]
    F. Bethuel, A characterization of maps in H1(B3,S2) which can be approximated by smooth maps, to appear.Google Scholar
  2. [BZ]
    F. Bethuel and X. Zheng, Density of smooth functions between two manifolds in Sobolev spaces, J. Funct. Analysis 80 (1988), 60–67.MathSciNetCrossRefMATHGoogle Scholar
  3. E] M. Escobedo, to appear.Google Scholar
  4. [EL]
    J. Eells and L. Lemaire, Bull. London Math. Soc. 10 (1978), 1–68.MathSciNetCrossRefMATHGoogle Scholar
  5. [SU1]
    R. Schoen and K. Uhlenbeck, A regularity theory for harmonic maps, J. Diff. Geom. 17 (1982), 307–335.MathSciNetMATHGoogle Scholar
  6. [SU2]
    R. Schoen and K. Uhlenbeck, Approximation theorems for Sobolev mappings, preprint.Google Scholar
  7. [W1]
    B. White, Infima of Energy Functions in homotopy classes, J. Diff. Geom. 23 (1986), 127–142.Google Scholar
  8. [W2]
    B. White, Homotopy classes in Sobolev spaces and the existence of energy minimizing maps, Acta Mathematica 160 (1988), 1–17.MathSciNetCrossRefGoogle Scholar

Copyright information

© Springer Science+Business Media New York 1990

Authors and Affiliations

  • Fabrice Bethuel
    • 1
  1. 1.CERMA ENPCLa CourtineFrance

Personalised recommendations