Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces

Abstract

We introduce an operator S on vector-valued maps u which has the ability to capture the relevant topological information carried by u. In particular, this operator is defined on maps that take values in a closed submanifold N of the Euclidean space R m , and coincides with the distributional Jacobian in case N is a sphere. More precisely, the range of S is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use S to characterise strong limits of smooth, N -valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivière. We also discuss applications to the study of manifold-valued maps of bounded variation. In a companion paper, we will consider applications to the asymptotic behaviour of minimisers of Ginzburg-Landau type functionals, with N -well potentials.

Improved Partial Regularity for Manifold-Constrained Minimisers of Subquadratic Energies

“…This "local density" result (Theorem 2.1) is obtained in Section 2. Similar results can be deduced from an earlier work by Pakzad and Rivière [35], but the statement we present here is more general and based on a different construction [14].…”

“…Let Ω ⊆ R d be a domain of dimension d ≥ k. Then, for any p ∈ [1, k) and any map u ∈ W 1,p (Ω, N ), there exists a relatively closed set Σ ⊆ Ω that satisfies the following properties: [38]. We are able to remove this technical restriction here because we appeal to the results in [14], which are based on a different construction than Pakzad and Rivière's one. However, the case k ≥ p is not covered by Theorem 2.1, because the construction carried over in [14] breaks down in this case.…”

“…We are able to remove this technical restriction here because we appeal to the results in [14], which are based on a different construction than Pakzad and Rivière's one. However, the case k ≥ p is not covered by Theorem 2.1, because the construction carried over in [14] breaks down in this case.…”

“…[8,9]; the counterpart for Besov spaces has been considered in [34]. In the setting of BV-spaces, we refer to [18,26] for the case N = S 1 , to [27] for the case N = RP q , and to [14] for compact manifolds with abelian fundamental group. If p ≥ 2, the same result holds true with Σ = ∅ [6,3].…”

Lifting for manifold-valued maps of bounded variation

“…In this paper, we consider the lifting problem when u is a BV-map. Previous works showed that the lifting problem for u ∈ BV(Ω, N ) has a positive answer in case N = S 1 (Giaquinta, Modica and Souček [18], Davila and Ignat [16], Ignat [24]), N = RP k (Bedford [4], Ignat and Lamy [25]) and more generally, if the fundamental group of N , π 1 (N ), is abelian [13]. The aim of this paper is to prove a lifting result for maps u ∈ BV(Ω, N ) without assuming that π 1 (N ) is abelian.…”