The approximation problem for Sobolev maps between two manifolds

The problem of density of smooth maps between two compact manifolds M n and N k was first considered by Eells and Lemaire ([EL]). If p>dimM", then WI'P~C ~ (by the Sobolev embedding theorem) and it is easy to see (using standard approximation methods) that C| ~, N k) is dense in WI"P(M ~, Nk). Schoen and Uhlenbeck [SU2], [SU3] have proved that smooth maps are dense in the limiting case p=dimM n. They also gave an example of non density of smooth maps: they showed that C~176 3, S 2) is not dense in H I ( B 3 , S 2 ) ; for instance the radial projection :r from B 3 to S 2 defined by :r(x)=x/lxl cannot be approximated by smooth maps. We consider in this paper two compact Riemannian manifolds M" and N k of dimension n and k respectively. N k is isometrically embedded in RI(/EN*). M n may have a boundary, but not N k. For l<~p<n, we consider the Sobolev space WI"P(M ~, N k) defined by: