Density of smooth maps in W k, p (M, N) for a close to critical domain dimension

Abstract

Assuming m −1 < kp < m, we prove that the space C ∞ (M, N ) of smooth mappings between compact Riemannian manifolds M, N (m = dim M) is dense in the Sobolev space W k, p (M, N ) if and only if π m−1 (N ) = {0}. If π m−1 (N ) = {0}, then every mapping in W k, p (M, N ) can still be approximated by mappings M → N which are smooth except in finitely many points.

Density of smooth maps for fractional Sobolev spaces W s,p into ℓ simply connected manifolds when s≥1

“…It seems that the topological condition π ⌊sp⌋ (N n ) = {0} is the only obstruction to the strong density of smooth maps in W s,p (Q m ; N n ). This is indeed true when s is an integer by a remarkable result of Bethuel [3,Theorem 1;16] for s = 1 which has been recently generalized by the authors [6,Theorem 4] for any s ∈ N (see also [13]):…”

Strong density for higher order Sobolev spaces into compact manifolds

“…The sketch of the proof we have announced in a previous work [6] for k = 2 and 2p > m − 1 is based on the strategy above but was organized differently following [46] (see also [20]). The opening technique was introduced by Brezis and Li [10] in their study of homotopy classes of W 1,p (Q m ; N n ).…”