Orateur
Description
In this talk, we introduce, via the Gagliardo completion, an extrapolation framework within De Giorgi’s Γ-convergence theory and develop its applications to variational problems arising in image processing, in particular, the Rudin-Osher-Fatemi (ROF) model. Special emphasis is given to new tools connecting extrapolation theory and variational analysis, specifically the extrapolation of compactness and the construction of variational K-functionals. These tools allow for the derivation of quantitative rates of convergence of minimizers. As an application, we resolve the inverse regularity problem in ROF: Regularity of the observed image that guarantees a prescribed rate of convergence to the solution of ROF is governed by the decay of variational K-functionals, which is explicitly captured by the Brezis-Van Schaftingen-Yung spaces. Moreover, this extrapolation framework provides a unified perspective on several limiting results, including the celebrated Bourgain-Brezis-Mironescu formula and its Γ-convergence counterparts due to Ponce.