Quantitative Γ-Convergence: A Variational Extrapolation Framework
Abstract
This paper introduces a new extrapolation framework within the Calculus of Variations. We develop a robust quantitative theory of the Γ-convergence approach of De Giorgi by combining it with Gagliardo's work on relative completions in interpolation theory. A central new tool is the extrapolation of \emph{variational K-functionals}, which associates to an arbitrary family of functionals a dynamic penalty method that simultaneously characterizes their Γ-limits and yields explicit rates of convergence for minimizers. As a technical foundation we prove compactness extrapolation theorems also involving Gagliardo completions. To illustrate the mechanism, we apply the technique to variational models arising in image processing, notably the Rudin--Osher--Fatemi (ROF) and Chan--Esedoglu models. Our main result in this setting is a contribution to the \emph{inverse regularity problem} associated to discrete approximations of the ROF model: we show that the required regularity of the observed image that guarantees a prescribed optimal rate of convergence is measured in terms of the decay of variational K-functionals. Surprisingly, our characterization is explicitly captured by using the recently introduced spaces of Brezis, Van Schaftingen, and Yung. Moreover, our methods unify, and considerably extend, classical pointwise convergence and Γ-versions of the Bourgain--Brezis--Mironescu and Maz'ya--Shaposhnikova formulae, as well as related results in a variety of settings that involve BMO-type functionals, heat kernels, Sobolev spaces on metric measure spaces, ... By bridging De Giorgi's variational convergence with Gagliardo's completion through variational K-functionals and extrapolation of compactness, this work establishes a quantitative approach to the Calculus of Variations.
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BibTeX
@misc{dominguez2026quantitative,
title = {{Quantitative $Γ$-Convergence: A Variational Extrapolation Framework}},
author = {Oscar Dominguez and Mario Milman},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.06645},
eprint = {2610.06645},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.06645},
note = {CC BY 4.0},
}
RIS
TY - UNPB
TI - Quantitative Γ-Convergence: A Variational Extrapolation Framework
AU - Oscar Dominguez
AU - Mario Milman
PY - 2026
DA - 2026/10/05
PB - arXiv
DO - 10.48550/arxiv.2610.06645
UR - https://arxiv.org/abs/2610.06645
AB - This paper introduces a new extrapolation framework within the Calculus of Variations. We develop a robust quantitative theory of the Γ-convergence approach of De Giorgi by combining it with Gagliardo's work on relative completions in interpolation theory. A central new tool is the extrapolation of \emph{variational K-functionals}, which associates to an arbitrary family of functionals a dynamic penalty method that simultaneously characterizes their Γ-limits and yields explicit rates of convergence for minimizers. As a technical foundation we prove compactness extrapolation theorems also involving Gagliardo completions. To illustrate the mechanism, we apply the technique to variational models arising in image processing, notably the Rudin--Osher--Fatemi (ROF) and Chan--Esedoglu models. Our main result in this setting is a contribution to the \emph{inverse regularity problem} associated to discrete approximations of the ROF model: we show that the required regularity of the observed image that guarantees a prescribed optimal rate of convergence is measured in terms of the decay of variational K-functionals. Surprisingly, our characterization is explicitly captured by using the recently introduced spaces of Brezis, Van Schaftingen, and Yung. Moreover, our methods unify, and considerably extend, classical pointwise convergence and Γ-versions of the Bourgain--Brezis--Mironescu and Maz'ya--Shaposhnikova formulae, as well as related results in a variety of settings that involve BMO-type functionals, heat kernels, Sobolev spaces on metric measure spaces, ... By bridging De Giorgi's variational convergence with Gagliardo's completion through variational K-functionals and extrapolation of compactness, this work establishes a quantitative approach to the Calculus of Variations.
ER -