Variational approximation of size-mass energies for k-dimensional currents^*

¹ CNRS, CMAP, École Polytechnique CNRS UMR 7641, Route de Saclay, 91128 Palaiseau Cedex, France.
² CMAP, École Polytechnique, CNRS UMR 7641, Route de Saclay, 91128 Palaiseau Cedex, France.
³ Laboratoire P. Painlevé, CNRS UMR 8524, Université Lille 1, 59655 Villeneuve d’Ascq Cedex, France.

^** Corresponding author: luca.ferrari@polytechnique.edu

Received: 21 October 2017
Accepted: 17 April 2018

Abstract

In this paper we produce a Γ-convergence result for a class of energies F_ε,a^k modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that F_ε,a¹ Γ-converges to a branched transportation energy whose cost per unit length is a function f_aⁿ⁻¹ depending on a parameter a > 0 and on the codimension n − 1. The limit cost f_a(m) is bounded from below by 1 + m so that the limit functional controls the mass and the length of the limit object. In the limit a ↓ 0 we recover the Steiner energy. We then generalize the approach to any dimension and codimension. The limit objects are now k-currents with prescribed boundary, the limit functional controls both their masses and sizes. In the limit a ↓ 0, we recover the Plateau energy defined on k-currents, k < n. The energies F_ε,a^k then could be used for the numerical treatment of the k-Plateau problem.