Non-local Functionals Related to the Total Variation and Connections with Image Processing

We present new results concerning the approximation of the total variation, $$\int _{\Omega } |\nabla u|$$∫Ω|∇u|, of a function u by non-local, non-convex functionals of the form $$\begin{aligned} \Lambda _\delta (u) = \int _{\Omega } \int _{\Omega } \frac{\delta \varphi \big ( |u(x) - u(y)|/ \delta \big )}{|x - y|^{d+1}} \, dx \, dy, \end{aligned}$$Λδ(u)=∫Ω∫Ωδφ(|u(x)-u(y)|/δ)|x-y|d+1dxdy,as $$\delta \rightarrow 0$$δ→0, where $$\Omega $$Ω is a domain in $$\mathbb {R}^d$$Rd and $$\varphi : [0, + \infty ) \rightarrow [0, + \infty )$$φ:[0,+∞)→[0,+∞) is a non-decreasing function satisfying some appropriate conditions. The mode of convergence is extremely delicate and numerous problems remain open. De Giorgi’s concept of $$\Gamma $$Γ-convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.