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arXiv · 1701.07627

Weak approximation by bounded Sobolev maps with values into complete manifolds

Pierre Bousquet·Augusto C. Ponce·Jean Van Schaftingen
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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.

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Pierre Bousquet, Augusto C. Ponce, Jean Van Schaftingen. 2017-01-26. Weak approximation by bounded Sobolev maps with values into complete manifolds. https://doi.org/10.1016/j.crma.2018.01.017

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