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

We continue the study of the space BV alpha(R-n) of functions with bounded fractional variation in R-n of order alpha is an element of (0, 1) introduced in our previous work (Comi and Stefani in J Funct Anal 277(10):3373-3435, 2019). After some technical improvements of certain results of Comi and Stefani (2019) which may be of some separated insterest, we deal with the asymptotic behavior of the fractional operators involved as alpha -> 1(-). We prove that the alpha-gradient of a W-1,W-P-function converges in L-p to the gradient for all p is an element of [1, +infinity) as alpha -> 1(-). Moreover, 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 -> 1(-). Finally, we prove that the fractional beta-variation converges to the fractional alpha-variation both pointwise and in the Gamma-limit sense as beta -> alpha(-) for any given alpha is an element of (0, 1).