Everywhere discontinuous harmonic maps into spheres

Let f~ be a bounded domain of R", S 2 the unit sphere of R 3 and (E, h) a surface homeomorphic to S 2 with a metric h. We may assume, using the Nash-Moser theorem, that E is isometrically imbedded in some Euclidean space R k. We consider the Sobolev space Hl(f~, ]E) = {u E Hi (n , R~): u(x) E ~, a.e. x E f~}. Let E(u)=fn IVu[ 2 be the Dirichlet energy for any u in Hl(f~, Z). For a sufficiently small neighborhood V of ~ in R k the projection lr of a point x of V is well defined. Weakly harmonic maps from f~ into ~ are critical points in Hi (n , Z) of the Dirichlet energy in the following way: