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

Abstract

We continue the study of the space $$BV^\alpha ({\mathbb {R}}^n)$$ 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 ) .

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

“…In [27,28], for a parameter α ∈ (0, 1), the third and fourth authors introduced the space of functions with bounded fractional variation…”

“…for all sufficiently regular functions f and vector fields ϕ. For an account on the existing literature related to these operators, we refer the reader to [13,14,27,28,41,60,[63][64][65][66][67][68][69][70] and to the references therein.…”

“…While the first paper [28] was focused on some geometric aspects of BV α functions, the subsequent work [27] was inspired by the celebrated Bourgain-Brezis-Mironescu formula [16] and the Γ-convergence result of Ambrosio-De Philippis-Martinazzi [3] and dealt with the asymptotic behavior of the fractional α-variation as α → 1 − . As already announced in [27], the main aim of present paper is to study the asymptotic behavior of the fractional α-variation as α → 0 + , in analogy with the asymptotic result of Maz ya-Shaposhnikova [49,50].…”

“…The asymptotic behavior of the fractional gradient ∇ α as α → 1 − was fully discussed in [27] (see also [14,Theorem 3.2] for a different proof of (10) below for the case p ∈ (1, +∞) via Fourier transform). Precisely, if f ∈ W 1,p (R n ) for some p ∈ [1, +∞), then f ∈ S α,p (R n ) for all α ∈ (0, 1) with…”

“…

We continue the study of the space BV α (R n ) of functions with bounded fractional variation in R n and of the distributional fractional Sobolev space S α,p (R n ), with p ∈ [1, +∞] and α ∈ (0, 1), considered in the previous works [27,28]. We first define the space BV 0 (R n ) and establish the identificationswhere H 1 (R n ) and L α,p (R n ) are the (real) Hardy space and the Bessel potential space, respectively.

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