On the weak limits of smooth maps for the Dirichlet energy between manifold

Abstract

We identify all the weak sequential limits of smooth maps in W 1;2 (M;N). In particular, this implies a necessary and su¢ cient topological condition for smooth maps to be weakly sequentially dense in W 1;2 (M;N). 1. Introduction AssumeM and N are smooth compact Riemannian manifolds without boundary and they are embedded into R and R respectively. The following spaces are of interest in the calculus of variations: W 1;2 (M;N) = n u 2W 1;2 M;R : u (x) 2 N a.e. x 2M o ; H W (M;N) = u 2W 1;2 (M;N) : there exists a sequence ui 2 C1 (M;N) such that ui * u in W 1;2 (M;N) : For a brief history and detailed references on the study of analytical and topological issues related to these spaces, one may refer to [2, 3, 7]. In particular, it follows from theorem 7.1 of [3] that a necessary condition for H W (M;N) = W 1;2 (M;N) is that M satis es the 1-extension property with respect to N (see section 2.2 of [3] for a de nition). It was conjectured in section 7 of [3] that the 1-extension property is also su¢ cient for H W (M;N) = W 1;2 (M;N). In [1, 7], it was shown that H W (M;N) = W 1;2 (M;N) when 1 (M) = 0 or 1 (N) = 0. Note that if 1 (M) = 0 or 1 (N) = 0, then M satis es the 1-extension property with respect to N . In section 8 of [4], it was proved that the above conjecture is true under the additional assumption that N satis es the 2-vanishing condition. The main aim of the present article is to con rm the conjecture in its full generality. More precisely, we have Theorem 1.1. Let M and N be smooth compact Riemannian manifolds without boundary (n 3). Take a Lipschitz triangulation h : K !M , then H W (M;N) = u 2W 1;2 (M;N) : u#;2 (h) has a continuous extension to M w.r.t. N = u 2W 1;2 (M;N) : u may be connected to some smooth maps : In addition, if 2 [M;N ] satis es hjjK1j = u#;2 (h), then we may nd a sequence of smooth maps ui 2 C1 (M;N) such that ui * u in W 1;2 (M;N), [ui] = and dui ! du a.e.. 1