Abstract
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. De Giorgi’s concept of Γ -convergence illuminates the situation, but also introduces mysterious novelties. The original motivation of our work comes from Image Processing.
| Original language | English (US) |
|---|---|
| Article number | 9 |
| Journal | Annals of PDE |
| Volume | 4 |
| Issue number | 1 |
| DOIs | |
| State | Published - Jun 1 2018 |
All Science Journal Classification (ASJC) codes
- Analysis
- Applied Mathematics
- Geometry and Topology
- Mathematical Physics
- Physics and Astronomy(all)
Keywords
- Bounded variation
- Non-convex functional
- Non-local functional
- Sobolev spaces
- Total variation
- Γ -Convergence