Weak approximation by bounded Sobolev maps with values into complete manifolds

Pierre Bousquet; Augusto C. Ponce; Jean Van Schaftingen

doi:10.1016/j.crma.2018.01.017

paper

Weak approximation by bounded Sobolev maps with values into complete manifolds

arXiv:1701.07627 · doi:10.1016/j.crma.2018.01.017

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 $p \in {1, \dots, m}$ . 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.