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Pré-Publication, Document De Travail Année : 2017

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

Résumé

In this paper we produce a Γ-convergence result for a class of energies F k ε,a modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that F 1 ε,a Γ-converges to a branched transportation energy whose cost per unit length is a function f n−1 a 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 k ε,a then can be used for the numerical treatment of the k-Plateau problem.
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Dates et versions

hal-01622540 , version 1 (24-10-2017)
hal-01622540 , version 2 (12-06-2018)
hal-01622540 , version 3 (05-12-2018)

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Antonin Chambolle, Luca Alberto Davide Ferrari, Benoît Merlet. Variational approximation of size-mass energies for k-dimensional currents. 2017. ⟨hal-01622540v2⟩
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