Analytical obstructions to the weak approximation of Sobolev mappings into manifolds

TL;DR

The article identifies analytical obstructions to the weak density of smooth Sobolev mappings into manifolds, showing that for every integer $p\ge 2$ there exists a compact target where $H^{1,p}_W(\mathcal M,\mathcal N) \subsetneq W^{1,p}(\mathcal M,\mathcal N)$ when $\dim \mathcal M>p$. It develops a robust framework built on bubbling, degree estimates, and relaxed energy growth, and leverages skeleton-to-manifold reductions to construct explicit obstructions; in particular, for $p=4n-1$ one can take $\mathcal N=\mathbb S^{2n}$ using Whitehead products with Hopf invariant $2$. The results extend to higher order Sobolev spaces $W^{s,p}$ with $s\ge 1$ and $sp\in\mathbb N$, establishing a broad class of obstructions, and they provide a systematic method to generate counterexamples that contrast analytical obstructions with classical topological ones. This work significantly advances the understanding of when weak approximation fails and offers new tools for analyzing singularities in nonlinear variational problems.

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 $.

Figures (4)

• Figure 1: On a generic square, a smooth map $u_k$ approximating $u$ should take at most points a value close to $u$ while engulfing at the other points all the singularities through the creation of bubbles.
• Figure 2: Given $u$ and $v$, the map $w$ is constructed so that it coincides with $v$ outside the larger balls, it is constant on the intermediate sphere and it is a rescaling of $v$ on the smaller balls; the map $w_0$ defined similarly outside the intermediate balls and constant inside those is homotopic to $u$.
• Figure 3: Thanks to an averaging argument, the spheres $\partial B_{\rho_j} (a_j)$ (in blue) can be chosen out of the growing balls generated by the balls $B_{\rho} (a)$ (in red) so that $u$ and $v$ have a small Sobolev energy and are at small average distance on them.
• Figure 4: The sets $Q_{\ell, 0}$, $G_{\ell, \square}$, and $G_{\ell, \gamma}$ for $\gamma = (-1, -1)$. The colored cubes on the boundary form the set $G_{\ell,\square}$, and the cubes that are further colored in blue form the set $G_{\ell,\gamma}$ for $\gamma = (-1,-1)$.

Theorems & Definitions (43)

• Theorem 1.1
• Theorem 1.2
• Theorem 1.3
• Theorem 1.4
• Proposition 2.1
• Proposition 2.2: Growing balls
• proof : Proof of Proposition \ref{['proposition_growing_balls']}
• Lemma 2.3: Merging balls
• proof
• Proposition 2.4