Non-convex, non-local functionals converging to the total variation

Brezis, Haïm; Nguyen, Hoai-Minh

doi:10.1016/j.crma.2016.11.002

Mathematical analysis/Partial differential equations

Non-convex, non-local functionals converging to the total variation
[Convergence de fonctionnelles non convexes et non locales vers la variation totale]

Présenté par : Haïm Brézis

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

Comptes Rendus. Mathématique, Tome 355 (2017) no. 1, pp. 24-27.

Résumé
Abstract

Nous présentons des résultats nouveaux concernant l'approximation de la variation totale $\int_{Ω} | \nabla u |$ d'une fonction u par des fonctionnelles non convexes et non locales de la forme

Λ_{δ} (u) = \int_{Ω} \int_{Ω} \frac{δ φ (| u (x) - u (y) | / δ)}{| x - y |^{d + 1}} d x d y,

quand

δ \to 0

, où Ω est un domaine de

R^{d}

φ : [0, + \infty) \to [0, + \infty)

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.

We present new results concerning the approximation of the total variation, $\int_{Ω} | \nabla u |$ , of a function u by non-local, non-convex functionals of the form

Λ_{δ} (u) = \int_{Ω} \int_{Ω} \frac{δ φ (| u (x) - u (y) | / δ)}{| x - y |^{d + 1}} d x d y,

δ \to 0

, where Ω is a domain in

R^{d}

and

φ : [0, + \infty) \to [0, + \infty)

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.

Reçu le : 2016-11-14
Accepté le : 2016-11-14
Publié le : 2016-12-12

DOI : 10.1016/j.crma.2016.11.002

Affiliations des auteurs :

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

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     title = {Non-convex, non-local functionals converging to the total variation},
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     pages = {24--27},
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Brezis, Haïm; Nguyen, Hoai-Minh. Non-convex, non-local functionals converging to the total variation. Comptes Rendus. Mathématique, Tome 355 (2017) no. 1, pp. 24-27. doi : 10.1016/j.crma.2016.11.002. http://www.numdam.org/articles/10.1016/j.crma.2016.11.002/

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