Volume 25, 2019
|Number of page(s)||39|
|Published online||20 September 2019|
Variational approximation of size-mass energies for k-dimensional currents*
CNRS, CMAP, École Polytechnique CNRS UMR 7641,
Route de Saclay,
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: email@example.com
Accepted: 17 April 2018
In this paper we produce a Γ-convergence result for a class of energies Fε,ak modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that Fε,a1 Γ-converges to a branched transportation energy whose cost per unit length is a function fan−1 depending on a parameter a > 0 and on the codimension n − 1. The limit cost fa(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 Fε,ak then could be used for the numerical treatment of the k-Plateau problem.
Mathematics Subject Classification: 49Q20 / 49J45 / 35A35
Key words: Γ-convergence / Steiner problem / plateau problem / phase-field approximations
© EDP Sciences, SMAI 2019
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