Uniform boundedness principles for Sobolev maps into manifolds

Abstract

Given a connected Riemannian manifold  , an -dimensional Riemannian manifold  which is either compact or the Euclidean space, ∈ [1, +∞) and ∈ (0, 1], we establish, for the problems of surjectivity of the trace, of weak-bounded approximation, of lifting and of superposition, that qualitative properties satisfied by every map in a nonlinear Sobolev space , (,  ) imply corresponding uniform quantitative bounds. This result is a nonlinear counterpart of the classical Banach-Steinhaus uniform boundedness principle in linear Banach spaces.The space, the energy and the topology on this space are independent of the embedding and can be defined intrinsically [30].1.1. Extension of traces. We first consider relationships between a qualitative and quantitative properties for the problem of surjectivity of the trace. In the setting of linear Sobolev spaces, given ∈ (0, 1), ∈ (1, +∞) and a manifold  which is either compact or the Euclidean space, the classical trace theory states that the restriction of continuous functions in ( × [0, +∞), ℝ) has a linear continuous extension to the trace operator tr ∶ +1∕ , ( × (0, +∞), ℝ) → , (, ℝ) and that the latter trace operator is surjective [1, Theorem 7.39;32, Chapter 10;61, Theorem 2.7.2]. A. Monteil is a postdoctoral researcher (chargé de recherches) by the Fonds de la Recherche Scientifique-FNRS; J. Van Schaftingen was supported by the Mandat d'Impulsion Scientifique F.4523.17, "Topological singularities of Sobolev maps" of the Fonds de la Recherche Scientifique-FNRS.

Minimal Ws,ns$W^{s,\frac{n}{s}}$‐harmonic maps in homotopy classes

“…To prove Theorem 4.1, we will construct a comparison map, the construction will be very similar to the one in the paper by Monteil–Van Schaftingen [84, Proof of Theorem 3.1]. We will be using the following lemmata from [84].…”

“…To prove Theorem 4.1, we will construct a comparison map, the construction will be very similar to the one in the paper by Monteil–Van Schaftingen [84, Proof of Theorem 3.1]. We will be using the following lemmata from [84]. The first lemma is called the opening of maps in the sense of Brezis–Li [19], and the purpose of it is to connect a given map continuously to a constant within the Sobolev space.…”

A fractional version of Rivière’s GL(n)-gauge

“…For readers convenience we present here a proof of a well-known result, which essentially says that in the critical Sobolev space a point has zero capacity. See for example [ 1 , Theorem 5.1.9]; compare also with a similar construction [ 23 , Lemma 3.2].…”

Minimal $W^{s,\frac{n}{s}}$-harmonic maps in homotopy classes

“…To prove Theorem 4.1 will construct a comparison map, the construction will be very similar to the one in the paper by Monteil-Van Schaftingen [79, Proof of Theorem 3.1]. We will be using the following lemmata from [79]. The first lemma, is called the opening of maps in the sense of Brezis-Li [20], and the purpose of it is to connect a given map continuously to a constant within the Sobolev space.…”