Calculus of Variations and Partial Differential Equations

, Volume 41, Issue 1, pp 45-69

First online:

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

  • Mariano GiaquintaAffiliated withScuola Normale Superiore Email author 
  • , Min-Chun HongAffiliated withDepartment of Mathematics, The University of Queensland
  • , Hao YinAffiliated withDepartment of Mathematics, The University of QueenslandDepartment of Mathematics, Shanghai Jiaotong University

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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)

58E20 58E50