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

Chambolle, Antonin; Ferrari, Luca A.D.; Merlet, Benoit

ESAIM: Control, Optimisation and Calculus of Variations, Tome 25 (2019), article no. 43

Suite au passage du modèle économique de la revue en S20, le texte intégral des articles des années 2019 et 2020 est accessible uniquement sur le site de l'éditeur et est réservé aux abonnés.

In this paper we produce a Γ-convergence result for a class of energies $$ modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that $$ Γ-converges to a branched transportation energy whose cost per unit length is a function $$ depending on a parameter a > 0 and on the codimension n − 1. The limit cost f$$(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 $$ then could be used for the numerical treatment of the k-Plateau problem.

Classification : 49Q20, 49J45, 35A35
Keywords: Γ-convergence, Steiner problem, plateau problem, phase-field approximations

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