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

Article history: Received 14 November 2016 Accepted 14 November 2016 Available online 12 December 2016 Presented by Haïm Brézis We present new results concerning the approximation of the total variation, ∫ |∇u|, of a function u by non-local, non-convex functionals of the form δ(u)= ∫ ∫ δφ (|u(x)− u(y)|/δ) |x− y|d+1 dxdy, as δ → 0, where 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. © 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). r é s u m é Nous présentons des résultats nouveaux concernant l’approximation de la variation totale ∫ |∇u| d’une fonction u par des fonctionnelles non convexes et non locales de la forme δ(u)= ∫ ∫ δφ (|u(x)− u(y)|/δ) |x− y|d+1 dxdy, E-mail addresses: [email protected] (H. Brezis), [email protected] (H.-M. Nguyen). http://dx.doi.org/10.1016/j.crma.2016.11.002 1631-073X/© 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/). H. Brezis, H.-M. Nguyen / C. R. Acad. Sci. Paris, Ser. I 355 (2017) 24–27 25 quand δ → 0, où 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. © 2016 Académie des sciences. Published by Elsevier Masson SAS. This is an open access article under the CC BY-NC-ND license (http://creativecommons.org/licenses/by-nc-nd/4.0/).