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

Abstract

In this paper we produce a $$\Gamma$$-convergence result for a class of energies $F k $\epsilon$,a$ modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that $F 1 $\epsilon$,a $\Gamma$$-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 $\downarrow$ 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 $\downarrow$ 0$, we recover the Plateau energy defined on k-currents, $k < n$. The energies $F k $\epsilon$,a$ then can be used for the numerical treatment of the k-Plateau problem.