Analytical obstructions to the weak approximation of Sobolev mappings into manifolds
Résumé
For any integer $ p \geq 2 $, we construct a compact Riemannian manifold $ \mathcal{N} $ , such that if $ \dim \mathcal{M} \geq p $, there is a map in the Sobolev space of mappings $ W^{1,p} (\mathcal{M}, \mathcal{N}) $ which is not a weak limit of smooth maps into $ \mathcal{N} $ due to a mechanism of analytical obstruction. For $ p = 4n -1 $, the target manifold can be taken to be the sphere $ \mathbb{S}^{2n} $ thanks to the construction by Whitehead product of maps with nontrivial Hopf invariant, generalizing the result by Bethuel for $ p = 4n -1 = 3 $. The results extend to higher order Sobolev spaces $ W^{s,p} $, with $ s \in \mathbb{R} $, $ s \geq 1 $, $ sp \in \mathbb{N} $, and $ sp \geq 2 $.
Origine | Fichiers produits par l'(les) auteur(s) |
---|