Analytical obstructions to the weak approximation of Sobolev mappings into manifolds

TL;DR

This paper constructs specific manifolds and maps demonstrating analytical obstructions to weak approximation in Sobolev spaces, extending known results to higher order spaces and particular dimensions.

Contribution

It introduces new examples of manifolds and maps where weak approximation fails in Sobolev spaces, generalizing previous results to higher order spaces and specific dimensions.

Findings

Existence of maps not approximable by smooth maps in Sobolev spaces.

Extension of obstructions to higher order Sobolev spaces.

Construction of target manifolds using Whitehead products and Hopf invariants.

Abstract

For any integer $ p \geq 2 $, we construct a compact Riemannian manifold $ \mathcal{N} $ such that if $ \dim \mathcal{M} > 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 \ge 2 $.