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

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Abstract

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 \(\mathscr {N}\) of the Euclidean space \(\mathbb {R}^m\), and coincides with the distributional Jacobian in case \(\mathscr {N}\) is a sphere. More precisely, 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, \(\mathscr {N}\)-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 \(\mathscr {N}\)-well potentials.

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Acknowledgements

The authors are grateful to the anonimous referees for their useful comments. G.C.’s research was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013)/ERC Grant Agreement No. 291053; by the Basque Government through the BERC 2018–2021 Program; by the Spanish Ministry of Science, Innovation and Universities: BCAM Severo Ochoa accreditation SEV-2017-0718; and by the Spanish Ministry of Economy and Competitiveness: MTM2017-82184-R. G. O. was partially supported by GNAMPA-INdAM.

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  1. Giacomo Canevari

    Present address: Dipartimento di Informatica, Università di Verona, Strada le Grazie 15, 37134, Verona, Italy

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  1. BCAM — Basque Center for Applied Mathematics, Alameda de Mazarredo 14, 48009, Bilbao, Spain

    Giacomo Canevari

  2. Dipartimento di Informatica, Università di Verona, Strada le Grazie 15, 37134, Verona, Italy

    Giandomenico Orlandi

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  1. Giacomo Canevari
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Correspondence to Giacomo Canevari.

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Canevari, G., Orlandi, G. Topological singular set of vector-valued maps, I: applications to manifold-constrained Sobolev and BV spaces. Calc. Var. 58, 72 (2019). https://doi.org/10.1007/s00526-019-1501-8

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