A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I

TL;DR

This paper investigates the asymptotic behavior of fractional variation spaces, showing convergence of fractional variations to classical and lower-order fractional variations as parameters approach their limits.

Contribution

It extends previous work by providing technical improvements and establishing convergence results for fractional variations as parameters tend to their limits.

Findings

Fractional $ ext{α}$-variation converges to De Giorgi's variation as $ ext{α} o 1^-$.

Fractional $eta$-variation converges to fractional $ ext{α}$-variation as $eta o ext{α}^-$.

Results hold both pointwise and in the $ ext{Γ}$-limit sense.

Abstract

We continue the study of the space $BV^\alpha(\mathbb{R}^n)$ of functions with bounded fractional variation in $\mathbb{R}^n$ of order $\alpha\in(0,1)$ introduced in arXiv:1809.08575, by dealing with the asymptotic behaviour of the fractional operators involved. After some technical improvements of certain results of our previous work, we prove that the fractional $\alpha$-variation converges to the standard De Giorgi's variation both pointwise and in the $\Gamma$-limit sense as $\alpha\to1^-$. We also prove that the fractional $\beta$-variation converges to the fractional $\alpha$-variation both pointwise and in the $\Gamma$-limit sense as $\beta\to\alpha^-$ for any given $\alpha\in(0,1)$.