Lifting in compact covering spaces for fractional Sobolev mappings

Mironescu, Petru; Van Schaftingen, Jean

doi:10.2140/apde.2021.14.1851

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Lifting in compact covering spaces for fractional Sobolev mappings

Petru Mironescu and Jean Van Schaftingen

Vol. 14 (2021), No. 6, 1851–1871

DOI: 10.2140/apde.2021.14.1851

Abstract

Let $π : \tilde{𝒩} \to 𝒩$ be a Riemannian covering, with $𝒩$ , $\tilde{𝒩}$ smooth compact connected Riemannian manifolds. If $ℳ$ is an $m$ -dimensional compact simply connected Riemannian manifold, $0 < s < 1$ and $2 \leq s p < m$ , we prove that every mapping $u \in W^{s, p} (ℳ, 𝒩)$ has a lifting in $W^{s, p}$ ; i.e., we have $u = π \circ ũ$ for some mapping $ũ \in W^{s, p} (ℳ, \tilde{𝒩})$ . Combined with previous contributions of Bourgain, Brezis and Mironescu and Bethuel and Chiron, our result settles completely the question of the lifting in Sobolev spaces over covering spaces.

The proof relies on an a priori estimate of the oscillations of $W^{s, p}$ maps with $0 < s < 1$ and $s p > 1$ , in dimension $1$ . Our argument also leads to the existence of a lifting when $0 < s < 1$ and $1 < s p < 2 \leq m$ , provided there is no topological obstruction on $u$ ; i.e., $u = π \circ ũ$ holds in this range provided $u$ is in the strong closure of $C^{\infty} (ℳ, 𝒩)$ .

However, when $0 < s < 1$ , $s p = 1$ and $m \geq 2$ , we show that an (analytical) obstruction still arises, even in the absence of topological obstructions. More specifically, we construct some map $u \in W^{s, p} (ℳ, 𝒩)$ in the strong closure of $C^{\infty} (ℳ, 𝒩)$ such that $u = π \circ ũ$ does not hold for any $ũ \in W^{s, p} (ℳ, \tilde{𝒩})$ .

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Keywords

analytical obstruction, finite-sheeted covering, Riemannian covering, fractional Sobolev spaces of mappings

Mathematical Subject Classification 2010

Primary: 46E35

Secondary: 58D15

Milestones

Received: 23 July 2019

Revised: 19 November 2019

Accepted: 19 March 2020

Published: 7 September 2021

Authors

Petru Mironescu
CNRS UMR 5208 Institut Camille Jordan Université Claude Bernard Lyon 1 Villeurbanne France	Simion Stoilow Institute of Mathematics of the Romanian Academy Bucureşti România

Jean Van Schaftingen
Institut de Recherche en Mathématique et Physique Université Catholique de Louvain Louvain-la-Neuve Belgium

Vol. 14, No. 6, 2021