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Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains

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Abstract

We define renormalised energies for maps that describe the first-order asymptotics of harmonic maps outside of singularities arising due to obstructions generated by the boundary data and the mutliple connectedness of the target manifold. The constructions generalise the definition by Bethuel et al. (Ginzburg–Landau vortices, progress in nonlinear differential equations and their applications, vol 13, Birkhäuser, Boston, 1994) for the circle. In general, the singularities are geometrical objects and the dependence on homotopic singularities can be studied through a new notion of synharmony. The renormalised energies are showed to be coercive and Lipschitz-continuous. The renormalised energies are associated to minimising renormalisable singular harmonic maps and minimising configurations of points can be characterised by the flux of the stress–energy tensor at the singularities. We compute the singular energy and the renormalised energy in several particular cases.

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Notes

  1. Such an embedding always exist in view of the Nash embedding theorem [36].

  2. Note that T could be a merging time for which \(A(T)\not \subset \Omega \), while \(A^-(T)\subset \Omega \) by construction.

  3. Note that the the flux of the stress–energy tensor through a small circle centered at the singularity \(a_i\) is independant of the (small) radius since the stress–energy tensor is divergence-free away from singularities.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Fry building, Woodland Road, Bristol, BS8 1UG, UK

    Antonin Monteil

  2. Laboratoire de Mathématiques d’Orsay, Université Paris-Saclay, CNRS, 91405, Orsay, France

    Rémy Rodiac

  3. Université catholique de Louvain, Institut de Recherche en Mathématique et Physique, Chemin du cyclotron 2, L7.01.02, 1348, Louvain-la-Neuve, Belgium

    Jean Van Schaftingen

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  1. Antonin Monteil
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Communicated by Y. Giga.

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A. Monteil was a postdoctoral researcher (chargé de recherches) by the Fonds de la Recherche Scientifique—FNRS over the period 2016–2019; R. Rodiac and J. Van Schaftingen were supported by the Mandat d’Impulsion Scientifique F.4523.17, “Topological singularities of Sobolev maps” of the Fonds de la Recherche Scientifique—FNRS: R.Rodiac was partially supported by the ANR project BLADE Jr. ANR-18-CE40-0023.

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Monteil, A., Rodiac, R. & Van Schaftingen, J. Renormalised energies and renormalisable singular harmonic maps into a compact manifold on planar domains. Math. Ann. 383, 1061–1125 (2022). https://doi.org/10.1007/s00208-021-02204-8

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