Weak approximation by bounded Sobolev maps with values into complete manifolds

摘要

We have recently introduced the trimming property for a complete Riemannian manifoldNnas a necessary and sufficient condition for bounded maps to be strongly dense inW1,p(Bm;Nn)whenp∈{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.