Weak approximation by bounded Sobolev maps with values into complete manifolds

Abstract We have recently introduced the trimming property for a complete Riemannian manifold N n as a necessary and sufficient condition for bounded maps to be strongly dense in W 1 , p ( B m ; N n ) when p ∈ { 1 , … , m } . We prove in this note that, even under a weaker notion of approximation, namely the weak sequential convergence, the trimming property remains necessary for the approximation in terms of bounded maps. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.