Calculus of Variations and Geometric Measure Theory
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G. Canevari - G. Orlandi

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

created by orlandi on 19 Dec 2018
modified by canevari on 29 Jan 2021


Accepted Paper

Inserted: 19 dec 2018
Last Updated: 29 jan 2021

Journal: Calc. Var. PDE
Pages: 45
Year: 2019
Doi: 10.1007/s00526-019-1501-8

ArXiv: 1712.10203 PDF


We introduce an operator $\mathbf{S}$ 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 $\mathbb{R}^m$, and coincides with the distributional Jacobian in case $N$ is a sphere. The range of $\mathbf{S}$ is a set of maps whose values are flat chains with coefficients in a suitable normed abelian group. In this paper, we use $\mathbf{S}$ to characterise strong limits of smooth, $N$-valued maps with respect to Sobolev norms, extending a result by Pakzad and Rivi\`ere. 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.

Keywords: Flat chains, Topological singularities, Density of smooth maps, lifting, manifold-valued maps

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