*Submitted Paper*

**Inserted:** 29 oct 2019

**Last Updated:** 10 may 2021

**Year:** 2019

**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 $\alpha\in(0,1)$.

**Keywords:**
Gamma-convergence, Fractional Gradient, fractional calculus, fractional perimeter, fractional derivative, fractional divergence, function with bounded fractional variation

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