A variational theory for integral functionals involving finite-horizon fractional gradients

TL;DR

This paper develops a variational framework for integral functionals involving truncated Riesz fractional gradients, establishing existence, asymptotic behavior, and localization results through novel translation operators and Gamma-convergence analysis.

Contribution

It introduces a new approach using translation operators to connect classical, fractional, and nonlocal gradients, advancing the existence theory and asymptotic analysis of these variational problems.

Findings

Quasiconvexity characterizes weak lower semicontinuity of nonlocal functionals.

Gamma-convergence results provide relaxation and homogenization insights.

Localization of fractional models to classical local models as fractional parameters vary.

Abstract

The center of interest in this work are variational problems with integral functionals depending on special nonlocal gradients. The latter correspond to truncated versions of the Riesz fractional gradient, as introduced in [Bellido, Cueto & Mora-Corral 2022] along with the underlying function spaces. We contribute several new aspects to both the existence theory of these problems and the study of their asymptotic behavior. Our overall proof strategy builds on finding suitable translation operators that allow to switch between the three types of gradients: classical, fractional, and nonlocal. These provide useful technical tools for transferring results from one setting to the other. Based on this approach, we show that quasiconvexity, which is the natural convexity notion in the classical -- and as shown in [Kreisbeck & Sch\"onberger 2022] also in the fractional -- calculus of variations, gives a necessary and sufficient condition for the weak lower semicontinuity of the nonlocal functionals as well. As a consequence of a general Gamma-convergence statement, we obtain relaxation and homogenization results. The analysis of the limiting behavior for varying fractional parameters yields, in particular, a rigorous localization with a classical local limit model.