## A. Chambolle - L. A. D. Ferrari - B. Merlet

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

created by ferrari on 15 Jan 2018

[

BibTeX]

*preprint*

**Inserted:** 15 jan 2018

**Year:** 2017

**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.