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

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 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 rightarrow 1^-. We prove that the alpha -gradient of a W^{1,p}-function converges in L^p to the gradient for all pin [1,+infty ) as alpha rightarrow 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 rightarrow 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 rightarrow alpha ^- for any given alpha in (0,1).