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

2017/12/29 by Giacomo Canevari, Canevari, Giacomo, Giandomenico Orlandi +1 · 1 citation
Mathematics · #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows

paper · pdf · doi:10.48550/arxiv.1712.10203

openalex publication_date 2017/12/29 · openalex created_date 2022/10/03 · openalex updated_date 2026/07/28

Abstract

We introduce an operator \S on vector-valued maps u which has the\nability to capture the relevant topological information carried by u. In\nparticular, this operator is defined on maps that take values in a closed\nsubmanifold N of the Euclidean space \ℝ^m, and coincides with the\ndistributional Jacobian in case N is a sphere. The range of \S is a\nset of maps whose values are flat chains with coefficients in a suitable normed\nabelian group. In this paper, we use \S to characterise strong limits\nof smooth, N-valued maps with respect to Sobolev norms, extending a result by\nPakzad and Rivi `ere. We also discuss applications to the study of\nmanifold-valued maps of bounded variation. In a companion paper, we will\nconsider applications to the asymptotic behaviour of minimisers of\nGinzburg-Landau type functionals.\n

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