Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks

Sisto Baldo; Van Phu Cuong Le; Annalisa Massaccesi; Giandomenico Orlandi; Sisto Baldo; Van Phu Cuong Le; Annalisa Massaccesi; Giandomenico Orlandi

doi:10.3934/mine.2023078

Mathematics in Engineering

2023, Volume 5, Issue 4: 1-19. doi: 10.3934/mine.2023078

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Research article

Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks

1.
Dipartimento di Informatica, Università di Verona, Italy
2.
Dipartimento di Tecnica e Gestione dei Sistemi Industriali, Università degli Studi di Padova, Italy

Received: 20 October 2022 Revised: 23 February 2023 Accepted: 23 February 2023 Published: 13 March 2023

We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g., Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb ^[10].
- liquid crystals,
- optimal mappings,
- Steiner tree problem,
- branched optimal transport,
- homological Plateau problem,
- $ G $-currents
Citation: Sisto Baldo, Van Phu Cuong Le, Annalisa Massaccesi, Giandomenico Orlandi. Energy minimizing maps with prescribed singularities and Gilbert-Steiner optimal networks[J]. Mathematics in Engineering, 2023, 5(4): 1-19. doi: 10.3934/mine.2023078

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Abstract

We investigate the relation between energy minimizing maps valued into spheres having topological singularities at given points and optimal networks connecting them (e.g., Steiner trees, Gilbert-Steiner irrigation networks). We show the equivalence of the corresponding variational problems, interpreting in particular the branched optimal transport problem as a homological Plateau problem for rectifiable currents with values in a suitable normed group. This generalizes the pioneering work by Brezis, Coron and Lieb ^[10].

References

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