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

created by stefani on 29 Oct 2019
modified on 20 Nov 2019

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Submitted Paper

Inserted: 29 oct 2019
Last Updated: 20 nov 2019

Year: 2019

ArXiv: 1910.13419 PDF

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 G. E. Comi and G. Stefani, "A distributional approach to fractional Sobolev spaces and fractional variation: existence of blow-up" in J. Funct. Anal. 277 (2019), no. 10, 3373–3435, 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 $\alpha\in(0,1)$.