Calculus of Variations and Geometric Measure Theory

S. Baldo - V. P. C. Le - A. Massaccesi - G. Orlandi

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

created by le on 23 Dec 2021
modified on 16 Mar 2023


Published Paper

Inserted: 23 dec 2021
Last Updated: 16 mar 2023

Journal: Mathematics in Engineering
Volume: 5
Year: 2023
Doi: 10.3934/mine.2023078


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.

Keywords: Irrigation problem, Steiner problem, Plateau problem, Calibration, Harmonic maps