Weak approximation by bounded Sobolev maps with values into complete manifolds (Q1704663)

The authors consider the density of bounded maps in the Sobolev space \(W^{1,p} (B^m,N^n)\) with respect to weak sequential convergence when \(p\in \{1,\ldots m\}\), \(B^m\) is the \(m\)-dimensional unit ball and \(N^n\) a connected closed embedded submanifold of \(\mathbb{R}^\nu\). The main result (Theorem 1.3) is that the so-called trimming property (introduced by the authors in [Ann. Mat. Pura Appl. (4) 196, No. 6, 2261--2301 (2017; Zbl 1380.58007)]) is a necessary condition for the density. The argument involves the construction of a Sobolev map having infinitely many analytical singularities going to infinity.