Calculus of Variations and Geometric Measure Theory

M. Bonafini - G. Orlandi - E. Oudet

Variational approximation of functionals defined on 1-dimensional connected sets: the planar case

created by orlandi on 12 Oct 2016
modified on 19 Jul 2019


Published Paper

Inserted: 12 oct 2016
Last Updated: 19 jul 2019

Journal: SIAM J. Math. Anal.
Volume: 50
Number: 6
Pages: 6307-6332
Year: 2018
Doi: 10.1137/17M1159452


In this paper we consider variational problems involving 1-dimensional connected sets in the euclidean plane, such as the classical Steiner tree problem and the irrigation (Gilbert-Steiner) problem. We relate them to optimal partition problems and provide a variational approximation through Modica-Mortola type energies proving a full $\Gamma$-convergence result. We also introduce a suitable convex relaxation and develop the corresponding numerical implementations. The proposed methods are quite general and the results we obtain can be extended to n-dimensional euclidean space or to more general manifold ambients, as shown in the companion paper 11.

Keywords: irrigation problem, Steiner problem, optimal partitions, Modica-Mortola