发表机构
CUNEF Universidad; Instituto Argentino de Matemática（CUNEF大学; 阿根廷数学研究所）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文提出一个变分外推框架，结合De Giorgi的$\Gamma$-收敛与Gagliardo完备化，通过变分$K$-泛函外推统一并扩展了多种收敛结果，并应用于图像处理模型，解决了ROF模型离散逼近的逆正则性问题。
AI 中文摘要
本文在变分学中引入了一个新的外推框架。我们将De Giorgi的$\Gamma$-收敛方法与其在插值理论中关于相对完备化的Gagliardo工作相结合，发展了一套稳健的定量理论。一个核心的新工具是变分$K$-泛函的外推，它将任意泛函族与一个动态惩罚方法联系起来，该方法同时刻画了它们的$\Gamma$-极限，并为极小化子提供了显式的收敛速率。作为技术基础，我们证明了涉及Gagliardo完备化的紧性外推定理。为了说明这一机制，我们将该技术应用于图像处理中出现的变分模型，特别是Rudin-Osher-Fatemi（ROF）和Chan-Esgedoglu模型。在此背景下，我们的主要结果是对ROF模型离散逼近相关的逆正则性问题的贡献：我们证明了保证预定最优收敛速率所需的观测图像的正则性，可以用变分$K$-泛函的衰减来衡量。令人惊讶的是，我们的刻画通过使用最近引入的Brezis、Van Schaftingen和Yung空间被明确地捕获。此外，我们的方法统一并显著扩展了Bourgain-Brezis-Mironescu和Maz'ya-Shaposhnikova公式的经典逐点收敛和$\Gamma$-版本，以及涉及BMO型泛函、热核、度量测度空间上的Sobolev空间等多种设置中的相关结果。通过将De Giorgi的变分收敛与Gagliardo的完备化通过变分$K$-泛函和紧性外推联系起来，这项工作为变分学建立了一种定量方法。
英文摘要
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.
Comments61 pages
