Calculus of Variations and Partial Differential Equations

, Volume 41, Issue 1, pp 45–69

A new approximation of relaxed energies for harmonic maps and the Faddeev model


    • Scuola Normale Superiore
  • Min-Chun Hong
    • Department of MathematicsThe University of Queensland
  • Hao Yin
    • Department of MathematicsThe University of Queensland
    • Department of MathematicsShanghai Jiaotong University

DOI: 10.1007/s00526-010-0353-z

Cite this article as:
Giaquinta, M., Hong, M. & Yin, H. Calc. Var. (2011) 41: 45. doi:10.1007/s00526-010-0353-z


We propose a new approximation for the relaxed energy E of the Dirichlet energy and prove that the minimizers of the approximating functionals converge to a minimizer u of the relaxed energy, and that u is partially regular without using the concept of Cartesian currents. We also use the same approximation method to study the variational problem of the relaxed energy for the Faddeev model and prove the existence of minimizers for the relaxed energy \({\tilde{E}_F}\) in the class of maps with Hopf degree ±1.

Mathematics Subject Classification (2000)


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© Springer-Verlag 2010