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

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].

Keywords

• liquid crystals
• optimal mappings
• steiner tree problem
• branched optimal transport
• homological plateau problem
• $ g $-currents