Non-convex, non-local functionals converging to the total variation
Haïm Brézis, Hoài-Minh Nguyên · Comptes Rendus Mathématique · 2016
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 Λ δ ( u ) = ∫ Ω ∫ Ω δ φ ( | u ( x ) − u ( y ) | / δ ) | x − y | d + 1 d x d y , as δ → 0 , where Ω is a domain in R d 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.