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

Chambolle, Antonin; Ferrari, Luca Alberto Davide; Merlet, Benoît

Document type :

Compte-rendu et recension critique d'ouvrage

DOI :

10.1051/cocv/2018027

Title :

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

Author(s) :

Chambolle, Antonin [Auteur]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Ferrari, Luca Alberto Davide [Auteur]
Centre de Mathématiques Appliquées de l'Ecole polytechnique [CMAP]
Merlet, Benoît [Auteur]

Reliable numerical approximations of dissipative systems [RAPSODI]
Laboratoire Paul Painlevé - UMR 8524 [LPP]

Journal title :

ESAIM: Control, Optimisation and Calculus of Variations

Pages :

Publisher :

EDP Sciences

Publication date :

2019-09-20

ISSN :

1292-8119

English keyword(s) :

Phase-field approximations
Gamma Convergence
Steiner Problem

HAL domain(s) :

Mathématiques [math]/Equations aux dérivées partielles [math.AP]

English abstract : [en]

In this paper we produce a $Γ$-convergence result for a class of energies $F k ε,a$ modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that $F 1 ε,a Γ$-converges to a branched transportation energy ...
Show more >In this paper we produce a $Γ$-convergence result for a class of energies $F k ε,a$ modeled on the Ambrosio-Tortorelli functional. For the choice k = 1 we show that $F 1 ε,a Γ$-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 ↓ 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 $F k ε,a$ then can be used for the numerical treatment of the k-Plateau problem.Show less >

Language :

Anglais

Popular science :

Non

ANR Project :

Théorie géométrique de la mesure et applications
Centre Européen pour les Mathématiques, la Physique et leurs Interactions

Collections :