The regularity problem for generalized harmonic maps into homogeneous spaces

Let be a Riemannian surface and be a standard sphere, or more generally a Riemannian manifold on which a Lie group,, acts transitively by isometries. We define generalized harmonic maps by extending the notion of weakly harmonic maps in a natural way (motivated by Noether's Theorem), to mapsu W _loc ^1,1 (, ). We prove that, under some slight technical restrictions, for 1 <-p < 2, there are generalized harmonic mapsu W ^1,p(, ) that are everywhere discontinuous (in particular, this solves an open problem proposed by F. Bethuel, H. Brezis and F. Hélein, in BBH]). We also show that the natural -regularity condition for such maps is to require <u to belong to the Lorentz space L(2, ). To prove this -regularity result we extend a compensated compactness result of R. Coifman, P.-L. Lions, Y. Meyer and S. Semmes, proved in CLMS], to the case of Lorentz spaces in duality.