Comptes Rendus
no. 1
Mathematical analysis/Partial differential equations
Non-convex, non-local functionals converging to the total variation

Presented by: Haïm Brézis

Haïm Brezis1, 2, 3; Hoai-Minh Nguyen4
1 Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
2 Department of Mathematics, Technion, Israel Institute of Technology, 32.000 Haifa, Israel
3 Laboratoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75252 Paris cedex 05, France
4 École polytechnique fédérale de Lausanne, EPFL, SB MATHAA CAMA, Station 8, CH-1015 Lausanne, Switzerland
Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 24-27.

We present new results concerning the approximation of the total variation, ∫Ω|∇u|, of a function u by non-local, non-convex functionals of the form

Λδ(u)=∫Ω∫Ωδφ(|u(x)−u(y)|/δ)|x−y|d+1dxdy,
as δ→0, where Ω is a domain in Rd and φ:[0,+∞)→[0,+∞) is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate, and numerous problems remain open. The original motivation of our work comes from Image Processing.

Nous présentons des résultats nouveaux concernant l'approximation de la variation totale ∫Ω|∇u| d'une fonction u par des fonctionnelles non convexes et non locales de la forme

Λδ(u)=∫Ω∫Ωδφ(|u(x)−u(y)|/δ)|x−y|d+1dxdy,
quand δ→0, où Ω est un domaine de Rd et φ:[0,+∞)→[0,+∞) est une fonction croissante vérifiant certaines hypothèses. Le mode de convergence est extrêmement délicat et de nombreux problèmes restent ouverts. La motivation provient du traitement d'images.

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2016.11.002

Haïm Brezis 1, 2, 3; Hoai-Minh Nguyen 4

1 Department of Mathematics, Hill Center, Busch Campus, 110 Frelinghuysen Road, Piscataway, NJ 08854, USA
2 Department of Mathematics, Technion, Israel Institute of Technology, 32.000 Haifa, Israel
3 Laboratoire Jacques-Louis-Lions, Université Pierre-et-Marie-Curie, 4, place Jussieu, 75252 Paris cedex 05, France
4 École polytechnique fédérale de Lausanne, EPFL, SB MATHAA CAMA, Station 8, CH-1015 Lausanne, Switzerland 
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Haïm Brezis; Hoai-Minh Nguyen. Non-convex, non-local functionals converging to the total variation. Comptes Rendus. Mathématique, Volume 355 (2017) no. 1, pp. 24-27. doi : 10.1016/j.crma.2016.11.002. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2016.11.002/

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