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

Abstract

AbstractWe continue the study of the space $$BV^\alpha ({\mathbb {R}}^n)$$ B V α ( R n ) of functions with bounded fractional variation in $${\mathbb {R}}^n$$ R n of order $$\alpha \in (0,1)$$ α ∈ ( 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 $$\alpha \rightarrow 1^-$$ α → 1 - . We prove that the $$\alpha $$ α -gradient of a $$W^{1,p}$$ W 1 , p -function converges in $$L^p$$ L p to the gradient for all $$p\in [1,+\infty )$$ p ∈ [ 1 , + ∞ ) as $$\alpha \rightarrow 1^-$$ α → 1 - . Moreover, we prove that the fractional $$\alpha $$ α -variation converges to the standard De Giorgi’s variation both pointwise and in the $$\Gamma $$ Γ -limit sense as $$\alpha \rightarrow 1^-$$ α → 1 - . Finally, we prove that the fractional $$\beta $$ β -variation converges to the fractional $$\alpha $$ α -variation both pointwise and in the $$\Gamma $$ Γ -limit sense as $$\beta \rightarrow \alpha ^-$$ β → α - for any given $$\alpha \in (0,1)$$ α ∈ ( 0 , 1 ) .