# Weak approximation by bounded Sobolev maps with values into complete manifolds

## TL;DR

This paper investigates the conditions under which bounded Sobolev maps into complete manifolds can be approximated by smoother maps, establishing the necessity of the trimming property even for weak convergence.

## Contribution

It extends the understanding of approximation properties in Sobolev spaces by showing the trimming property is necessary for weak sequential approximation.

## Key findings

- The trimming property is necessary for weak approximation by bounded Sobolev maps.
- Construction of Sobolev maps with infinitely many singularities demonstrates the necessity.
- Weak approximation results depend critically on the geometric property of the target manifold.

## 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 $p \in \{1, \dotsc, m\}$. 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.