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.

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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.

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