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Journal Articles ESAIM: Control, Optimisation and Calculus of Variations Year : 2019

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

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Abstract

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 and 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. ESAIM: Control, Optimisation and Calculus of Variations, 2019, 25 (2019) (43), pp.39. ⟨10.1051/cocv/2018027⟩. ⟨hal-01622540v3⟩
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