Density in W^s,p(Ω;N)

Let Ω be a smooth bounded domain in Rn, 0<s<∞ and 1≤p<∞. We prove that C∞(Ω[U+203E];S1) is dense in Ws,p(Ω;S1) except when 1≤sp<2 and n≥2. The main ingredient is a new approximation method for W^s,p-maps when s<1. With 0<s<1, 1≤p<∞ and sp<n, Ω a ball, and N a general compact connected manifold, we prove that C∞(Ω[U+203E];N) is dense in W^s,p(Ω;N) if and only if π_[sp](N)=0. This supplements analogous results obtained by Bethuel when s=1, and by Bousquet, Ponce and Van Schaftingen when s=2, 3, . . . . [General domains Ω have been treated by Hang and Lin when s=1; our approach allows to extend their result to s<1.] The case where s>1, s∉N, is still open.