Inserted: 12 oct 2016
Last Updated: 14 sep 2017
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