Abstract: In this talk, we introduce, via the Gagliardo completion, an extrapolation framework within the theory of $\mathrm{\Gamma }$-convergence 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: The sharp 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 framework unifies and extend several limiting formulas, including the $\mathrm{\Gamma }$-convergence counterparts of the Bourgain—Brezis—Mironescu and Maz’ya—Shaposhnikova formulas, extensions to metric measure spaces, BMO-type functionals, heat-type energies, …

 

This talk is based on joint work with M. Milman.

 

Link Information: Zoom meeting https://us02web.zoom.us/j/87303952902 

 

Organizer: Daniel Spector (NTNU)