Calculus of Variations and Geometric Measure Theory
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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



Inserted: 15 jan 2018

Year: 2017

ArXiv: 1710.08808 PDF


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.

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