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 \(p\in [1,+\infty )\) 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 \(\alpha \in (0,1)\).
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1 Introduction
1.1 A distributional approach to fractional variation
In our previous work [27], we introduced the space \(BV^\alpha ({\mathbb {R}}^n)\) of functions with bounded fractional variation in \({\mathbb {R}}^n\) of order \(\alpha \in (0,1)\). Precisely, a function \(f\in L^1({\mathbb {R}}^n)\) belongs to the space \(BV^\alpha ({\mathbb {R}}^n)\) if its fractional \(\alpha \)-variation
is finite. Here
is the fractional \(\alpha \)-divergence of \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), where
for any given \(\alpha \in (0,1)\). The operator \(\mathrm {div}^\alpha \) was introduced in [72] as the natural dual operator of the much more studied fractional \(\alpha \)-gradient
defined for all \(f\in C^\infty _c({\mathbb {R}}^n)\). For an account on the existing literature on the operator \(\nabla ^\alpha \), see [68, Section 1]. Here we only refer to [66,67,68,69,70, 72,73,74] for the articles tightly connected to the present work and to [63, Section 15.2] for an agile presentation of the fractional operators defined in (1.2) and in (1.4) and of some of their elementary properties. According to [70, Section 1], it is interesting to notice that [42] seems to be the earliest reference for the operator defined in (1.4).
The operators in (1.2) and in (1.4) are dual in the sense that
for all \(f\in C^\infty _c({\mathbb {R}}^n)\) and \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), see [72, Section 6] and [27, Lemma 2.5]. Moreover, both operators have good integrability properties when applied to test functions, namely \(\nabla ^\alpha f\in L^p({\mathbb {R}}^n)\) and \(\mathrm {div}^\alpha \varphi \in L^p({\mathbb {R}}^n;{\mathbb {R}}^n)\) for all \(p\in [1,+\infty ]\) for any given \(f\in C^\infty _c({\mathbb {R}}^n)\) and \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), see [27, Corollary 2.3].
The integration-by-part formula (1.5) represents the starting point for the distributional approach to fractional Sobolev spaces and the fractional variation we developed in [27]. In fact, similarly to the classical case, a function \(f\in L^1({\mathbb {R}}^n)\) belongs to \(BV^\alpha ({\mathbb {R}}^n)\) if and only if there exists a finite vector-valued Radon measure \(D^{\alpha } f \in {\mathscr {M}}({\mathbb {R}}^n; {\mathbb {R}}^{n})\) such that
for all \(\varphi \in C^{\infty }_{c}({\mathbb {R}}^n; {\mathbb {R}}^{n})\), see [27, Theorem 3.2].
Motivated by (1.6) and similarly to the classical case, we can define the weak fractional \(\alpha \)-gradient of a function \(f\in L^p({\mathbb {R}}^n)\), with \(p\in [1,+\infty ]\), as the function \(\nabla ^\alpha _w f\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n;{\mathbb {R}}^n)\) satisfying
for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\). For \(\alpha \in (0,1)\) and \(p\in [1,+\infty ]\), we can thus define the distributional fractional Sobolev space
naturally endowed with the norm
It is interesting to compare the distributional fractional Sobolev spaces \(S^{\alpha ,p}({\mathbb {R}}^n)\) with the well-known fractional Sobolev space \(W^{\alpha ,p}({\mathbb {R}}^n)\), that is, the space
endowed with the norm
If \(p=+\infty \), then \(W^{\alpha ,\infty }({\mathbb {R}}^n)\) naturally coincides with the space of bounded \(\alpha \)-Hölder continuous functions endowed with the usual norm (see [32] for a detailed account on the spaces \(W^{\alpha ,p}\)).
For the case \(p=1\), starting from the very definition of the fractional gradient \(\nabla ^\alpha \), it is plain to see that \(W^{\alpha ,1}({\mathbb {R}}^n)\subset S^{\alpha ,1}({\mathbb {R}}^n)\subset BV^\alpha ({\mathbb {R}}^n)\) with both (strict) continuous embeddings, see [27, Theorems 3.18 and 3.25].
For the case \(p\in (1,+\infty )\), instead, it is known that \(S^{\alpha ,p}({\mathbb {R}}^n)\supset L^{\alpha ,p}({\mathbb {R}}^n)\) with continuous embedding, where \(L^{\alpha ,p}({\mathbb {R}}^n)\) is the Bessel potential space of parameters \(\alpha \in (0,1)\) and \(p\in (1,+\infty )\), see [27, Section 3.9] and the references therein. In the subsequent paper [26], it will be proved that also the inclusion \(S^{\alpha ,p}({\mathbb {R}}^n)\subset L^{\alpha ,p}({\mathbb {R}}^n)\) holds continuously, so that the spaces \(S^{\alpha ,p}({\mathbb {R}}^n)\) and \(L^{\alpha ,p}({\mathbb {R}}^n)\) coincide. In particular, we get the following relations: \(S^{\alpha +\varepsilon ,p}({\mathbb {R}}^n)\subset W^{\alpha , p}({\mathbb {R}}^{n})\subset S^{\alpha -\varepsilon ,p}({\mathbb {R}}^n)\) with continuous embeddings for all \(\alpha \in (0,1)\), \(p\in (1,+\infty )\) and \(0<\varepsilon <\min \{\alpha ,1-\alpha \}\), see [69, Theorem 2.2]; \(S^{\alpha , 2}({\mathbb {R}}^{n})=W^{\alpha , 2}({\mathbb {R}}^{n})\) for all \(\alpha \in (0, 1)\), see [69, Theorem 2.2]; \(W^{\alpha ,p}({\mathbb {R}}^n)\subset S^{\alpha ,p}({\mathbb {R}}^n)\) with continuous embedding for all \(\alpha \in (0,1)\) and \(p\in (1,2]\), see [76, Chapter V, Section 5.3].
In the geometric regime \(p=1\), our distributional approach to the fractional variation naturally provides a new definition of distributional fractional perimeter. Precisely, for any open set \(\Omega \subset {\mathbb {R}}^n\), the fractional Caccioppoli \(\alpha \)-perimeter in \(\Omega \) of a measurable set \(E\subset {\mathbb {R}}^n\) is the fractional \(\alpha \)-variation of \(\chi _E\) in \(\Omega \), i.e.
Thus, E is a set with finite fractional Caccioppoli \(\alpha \)-perimeter in \(\Omega \) if \(|D^\alpha \chi _E|(\Omega )<+\infty \).
Similarly to the aforementioned embedding \(W^{\alpha ,1}({\mathbb {R}}^n)\subset BV^\alpha ({\mathbb {R}}^n)\), we have the inequality
for any open set \(\Omega \subset {\mathbb {R}}^n\), see [27, Proposition 4.8], where
is the standard fractional \(\alpha \)-perimeter of a measurable set \(E\subset {\mathbb {R}}^n\) relative to the open set \(\Omega \subset {\mathbb {R}}^n\) (see [28] for an account on the fractional perimeter \(P_\alpha \)). Note that, by definition, the fractional \(\alpha \)-perimeter of E in \({\mathbb {R}}^n\) is simply \(P_\alpha (E):=P_\alpha (E;{\mathbb {R}}^n)=[\chi _E]_{W^{\alpha ,1}({\mathbb {R}}^n)}\). We remark that inequality (1.9) is strict in most of the cases, as shown in Sect. 2.6 below. This completely answers a question left open in our previous work [27].
1.2 Asymptotics and \(\Gamma \)-convergence in the standard fractional setting
The fractional Sobolev space \(W^{\alpha ,p}({\mathbb {R}}^n)\) can be understood as an ‘intermediate space’ between the space \(L^p({\mathbb {R}}^n)\) and the standard Sobolev space \(W^{1,p}({\mathbb {R}}^n)\). In fact, \(W^{\alpha ,p}({\mathbb {R}}^n)\) can be recovered as a suitable (real) interpolation space between the spaces \(L^p({\mathbb {R}}^n)\) and \(W^{1,p}({\mathbb {R}}^n)\). We refer to [13, 78] for a general introduction on interpolation spaces and to [54] for a more specific treatment of the interpolation space between \(L^p({\mathbb {R}}^n)\) and \(W^{1,p}({\mathbb {R}}^n)\).
One then naturally expects that, for a sufficiently regular function f, the fractional Sobolev seminorm \([f]_{W^{\alpha ,p}({\mathbb {R}}^n)}\), multiplied by a suitable renormalising constant, should tend to \(\Vert f\Vert _{L^p({\mathbb {R}}^n)}\) as \(\alpha \rightarrow 0^+\) and to \(\Vert \nabla f\Vert _{L^p({\mathbb {R}}^n)}\) as \(\alpha \rightarrow 1^-\). Indeed, for \(p\in [1,+\infty )\), it is known that
for all \(f\in \bigcup _{\alpha \in (0,1)}W^{\alpha ,p}({\mathbb {R}}^n)\), while
for all \(f\in W^{1,p}({\mathbb {R}}^n)\). Here \(A_{n,p},B_{n,p}>0\) are two constants depending only on n, p. The limit (1.11) was proved in [51, 52], while the limit (1.12) was established in [14]. As proved in [30], when \(p=1\) the limit (1.12) holds in the more general case of BV functions, that is,
for all \(f\in BV({\mathbb {R}}^n)\). For a different approach to the limits in (1.11) and in (1.13) based on interpolation techniques, see [54].
The limits (1.12) and (1.13) are special consequences of the celebrated Bourgain–Brezis–Mironescu (BBM, for short) formula
where \(C_{n,p}>0\) is a constant depending only on n and p, and \((\varrho _k)_{k\in {\mathbb {N}}}\subset L^1_{{{\,\mathrm{loc}\,}}}([0,+\infty ))\) is a sequence of non-negative radial mollifiers such that
The BBM formula (1.14) has stimulated a profound development in the asymptotic analysis in the fractional framework. On the one hand, the limit (1.14) played a central role in several applications, such as Brezis’ analysis [18] on how to recognize constant functions, innovative characterizations of Sobolev and BV functions and \(\Gamma \)-convergence results [6,7,8, 11, 16, 48,49,50, 56,57,59, 63], approximation of Sobolev norms and image processing [20, 22,23,24], and last but not least fractional Hardy and Poincaré inequalities [15, 38, 61]. On the other hand, the BBM formula (1.14) has suggested an alternative path to fractional asymptotic analysis by means of interpolation techniques [54, 65]. Recently, the BBM formula in (1.14) has been revisited in terms of a.e. pointwise convergence [21] and in connection with weak \(L^p\) quasi-norms [25], where the now-called Brezis–Van Schaftingen–Yung space
defined for \(\alpha \in (0,1]\) and \(p\in [1,+\infty )\), has opened a very promising perspective in the field [33].
The limits (1.11)–(1.14) have been connected to variational problems [10], generalized to various function spaces, for example Besov spaces [43, 79], Orlicz spaces [2, 36, 37] and magnetic and anisotropic Sobolev spaces [45, 58,59,60, 75], and extended to various ambient spaces, like compact connected Riemannian manifolds [44], the flat torus [5], Carnot groups [12, 49] and complete doubling metric-measure spaces supporting a local Poincaré inequality [31].
Concerning the fractional perimeter \(P_\alpha \) given in (1.10), one has some additional information besides equations (1.11) and (1.13).
On the one hand, thanks to [64, Theorem 1.2], the fractional \(\alpha \)-perimeter \(P_\alpha \) enjoys the following fractional analogue of Gustin’s Boxing Inequality (see [41] and [35, Corollary 4.5.4]): there exists a dimensional constant \(c_n>0\) such that, for any bounded open set \(E\subset {\mathbb {R}}^n\), one can find a covering
of open balls such that
Inequality (1.15) bridges the two limiting behaviors given by (1.11) and (1.13) and provides a useful tool for recovering Gagliardo–Nirenberg–Sobolev and Poincaré–Sobolev inequalities that remain stable as the exponent \(\alpha \in (0,1)\) approaches the endpoints.
On the other hand, by [3, Theorem 2], the fractional \(\alpha \)-perimeter \(P_\alpha \) \(\Gamma \)-converges in \(L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) to the standard De Giorgi’s perimeter P as \(\alpha \rightarrow 1^-\), that is, if \(\Omega \subset {\mathbb {R}}^n\) is a bounded open set with Lipschitz boundary, then
for all measurable sets \(E\subset {\mathbb {R}}^n\), where \(\omega _{n}\) is the volume of the unit ball in \({\mathbb {R}}^{n}\) (it should be noted that in [3] the authors use a slightly different definition of the fractional \(\alpha \)-perimeter, since they consider the functional \({\mathcal {J}}_{\alpha }(E, \Omega ) := \frac{1}{2} P_{\alpha }(E, \Omega )\)). For a complete account on \(\Gamma \)-convergence, we refer the reader to the monographs [17, 29] (throughout all the paper, with the symbol \(\Gamma (X)\text { -}\lim \) we denote the \(\Gamma \)-convergence in the ambient metric space X). The convergence in (1.16), besides giving a \(\Gamma \)-convergence analogue of the limit in (1.13), is tightly connected with the study of the regularity properties of non-local minimal surfaces, that is, (local) minimisers of the fractional \(\alpha \)-perimeter \(P_\alpha \).
1.3 Asymptotics and \(\Gamma \)-convergence for the fractional \(\alpha \)-variation as \(\alpha \rightarrow 1^-\)
The main aim of the present work is to study the asymptotic behavior of the fractional \(\alpha \)-variation (1.1) as \(\alpha \rightarrow 1^-\), both in the pointwise and in the \(\Gamma \)-convergence sense.
We provide counterparts of the limits (1.12) and (1.13) for the fractional \(\alpha \)-variation. Indeed, we prove that, if \(f\in W^{1,p}({\mathbb {R}}^n)\) for some \(p\in [1,+\infty )\), then \(f\in S^{\alpha ,p}({\mathbb {R}}^n)\) for all \(\alpha \in (0,1)\) and, moreover,
In the geometric regime \(p=1\), we show that if \(f\in BV({\mathbb {R}}^n)\) then \(f\in BV^\alpha ({\mathbb {R}}^n)\) for all \(\alpha \in (0,1)\) and, in addition,
and
We are also able to treat the case \(p=+\infty \). In fact, we prove that if \(f\in W^{1,\infty }({\mathbb {R}}^n)\) then \(f\in S^{\alpha ,\infty }({\mathbb {R}}^n)\) for all \(\alpha \in (0,1)\) and, moreover,
and
We refer the reader to Theorem 4.9, Theorem 4.11 and Theorem 4.12 below for the precise statements. We warn the reader that the symbol ‘\(\rightharpoonup \)’ appearing in (1.18) and (1.20) denotes the weak*-convergence, see Sect. 2.1 below for the notation.
Some of the above results were partially announced in [71]. In a similar perspective, we also refer to the work [53], where the authors proved convergence results for non-local gradient operators on BV functions defined on bounded open sets with smooth boundary. The approach developed in [53] is however completely different from the asymptotic analysis we presently perform for the fractional operator defined in (1.4), since the boundedness of the domain of definition of the integral operators considered in [53] plays a crucial role.
Notice that the renormalising factor \((1-\alpha )^\frac{1}{p}\) is not needed in the limits (1.17)–(1.21), contrarily to what happened for the limits (1.12) and (1.13). In fact, this difference should not come as a surprise, since the constant \(\mu _{n,\alpha }\) in (1.3), encoded in the definition of the operator \(\nabla ^\alpha \), satisfies
and thus plays a similar role of the factor \((1-\alpha )^\frac{1}{p}\) in the limit as \(\alpha \rightarrow 1^-\). Thus, differently from our previous work [27], the constant \(\mu _{n,\alpha }\) appearing in the definition of the operators \(\nabla ^\alpha \) and \(\mathrm {div}^\alpha \) is of crucial importance in the asymptotic analysis developed in the present paper.
Another relevant aspect of our approach is that convergence as \(\alpha \rightarrow 1^-\) holds true not only for the total energies, but also at the level of differential operators, in the strong sense when \(p\in (1,+\infty )\) and in the weak* sense for \(p=1\) and \(p=+\infty \). In simpler terms, the non-local fractional \(\alpha \)-gradient \(\nabla ^\alpha \) converges to the local gradient \(\nabla \) as \(\alpha \rightarrow 1^-\) in the most natural way every time the limit is well defined.
We also provide a counterpart of (1.16) for the fractional \(\alpha \)-variation as \(\alpha \rightarrow 1^-\). Precisely, we prove that, if \(\Omega \subset {\mathbb {R}}^n\) is a bounded open set with Lipschitz boundary, then
for all measurable set \(E\subset {\mathbb {R}}^n\), see Theorem 4.16. In view of (1.9), one may ask whether the \(\Gamma \text { -}\limsup \) inequality in (1.23) could be deduced from the \(\Gamma \text { -}\limsup \) inequality in (1.16). In fact, by employing (1.9) together with (1.16) and (1.22), one can estimate
However, we have \(\frac{2\omega _{n - 1}}{\omega _{n}} > 1\) for any \(n \ge 2\) and thus the \(\Gamma \text { -}\limsup \) inequality in (1.23) follows from the \(\Gamma \text { -}\limsup \) inequality in (1.16) only in the case \(n=1\). In a similar way, one sees that the \(\Gamma \text { -}\liminf \) inequality in (1.23) implies the \(\Gamma \text { -}\liminf \) inequality in (1.16) only in the case \(n=1\).
Besides the counterpart of (1.16), our approach allows to prove that \(\Gamma \)-convergence holds true also at the level of functions. Indeed, if \(f\in BV({\mathbb {R}}^n)\) and \(\Omega \subset {\mathbb {R}}^n\) is an open set such that either \(\Omega \) is bounded with Lipschitz boundary or \(\Omega ={\mathbb {R}}^n\), then
One can regard the limit (1.24) as an analogue of the \(\Gamma \)-convergence results known in the usual fractional setting, see [57, 62]. We refer the reader to Theorems 4.13, 4.14 and 4.17 for the (even more general) results in this direction. Again, as before and thanks to the asymptotic behavior (1.22), the renormalising factor \((1-\alpha )\) is not needed in the limits (1.23) and (1.24).
As a byproduct of the techniques developed for the asymptotic study of the fractional \(\alpha \)-variation as \(\alpha \rightarrow 1^-\), we are also able to characterize the behavior of the fractional \(\beta \)-variation as \(\beta \rightarrow \alpha ^-\), for any given \(\alpha \in (0,1)\). On the one hand, if \(f\in BV^\alpha ({\mathbb {R}}^n)\), then
and, moreover,
see Theorem 5.4. On the other hand, if \(f\in BV^\alpha ({\mathbb {R}}^n)\) and \(\Omega \subset {\mathbb {R}}^n\) is an open set such that either \(\Omega \) is bounded and \(|D^\alpha f|(\partial \Omega )=0\) or \(\Omega ={\mathbb {R}}^n\), then
1.4 Future developments: asymptotics for the fractional \(\alpha \)-variation as \(\alpha \rightarrow 0^+\)
Having in mind the limit (1.11), it would be interesting to understand what happens to the fractional \(\alpha \)-variation (1.1) as \(\alpha \rightarrow 0^+\). Note that
so there is no renormalization factor as \(\alpha \rightarrow 0^+\), differently from (1.22).
At least formally, as \(\alpha \rightarrow 0^+\) the fractional \(\alpha \)-gradient in (1.4) is converging to the operator
The operator in (1.26) is well defined (in the principal value sense) for all \(f\in C^\infty _c({\mathbb {R}}^n)\) and, actually, coincides with the well-known vector-valued Riesz transform Rf, see [39, Section 5.1.4] and [76, Chapter 3]. Similarly, the fractional \(\alpha \)-divergence in (1.2) is formally converging to the operator
which is well defined (in the principal value sense) for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\).
In perfect analogy with what we did before, we can introduce the space \(BV^0({\mathbb {R}}^n)\) as the space of functions \(f\in L^1({\mathbb {R}}^n)\) such that the quantity
is finite. Surprisingly (and differently from the fractional \(\alpha \)-variation, recall [27, Section 3.10]), it turns out that \(|D^0 f|\ll {\mathscr {L}}^{n}\) for all \(f\in BV^0({\mathbb {R}}^n)\). More precisely, one can actually prove that \(BV^0({\mathbb {R}}^n)=H^1({\mathbb {R}}^n)\), in the sense that \(f\in BV^0({\mathbb {R}}^n)\) if and only if \(f\in H^1({\mathbb {R}}^n)\), with \(D^0f=Rf{\mathscr {L}}^{n}\) in \({\mathscr {M}}({\mathbb {R}}^n;{\mathbb {R}}^n)\). Here
is the (real) Hardy space, see [77, Chapter III] for the precise definition. Thus, it would be interesting to understand for which functions \(f\in L^1({\mathbb {R}}^n)\) the fractional \(\alpha \)-gradient \(\nabla ^\alpha f\) tends (in a suitable sense) to the Riesz transform Rf as \(\alpha \rightarrow 0^+\). Of course, if \(Rf\notin L^1({\mathbb {R}}^n;{\mathbb {R}}^n)\), that is, \(f\notin H^1({\mathbb {R}}^n)\), then one cannot expect strong convergence in \(L^1\) and, instead, may consider the asymptotic behavior of the rescaled fractional gradient \(\alpha \,\nabla ^\alpha f\) as \(\alpha \rightarrow 0^+\), in analogy with the limit in (1.11). This line of research, as well as the identifications \(BV^0=H^1\) and \(S^{\alpha ,p}=L^{\alpha ,p}\) mentioned above, it is the subject of the subsequent paper [26].
1.5 Organization of the paper
The paper is organized as follows.
In Sect. 2, after having briefly recalled the definitions and the main properties of the operators \(\nabla ^\alpha \) and \(\mathrm {div}^\alpha \), we extend certain technical results of [27].
In Sect. 3, we prove several integrability properties of the fractional \(\alpha \)-gradient and two useful representation formulas for the fractional \(\alpha \)-variation of functions with bounded De Giorgi’s variation. We are also able to prove similar results for the fractional \(\beta \)-gradient of functions with bounded fractional \(\alpha \)-variation, see Sect. 3.4.
In Sect. 4, we study the asymptotic behavior of the fractional \(\alpha \)-variation as \(\alpha \rightarrow 1^-\) and prove pointwise-convergence and \(\Gamma \)-convergence results, dealing separately with the integrability exponents \(p=1\), \(p\in (1,+\infty )\) and \(p=+\infty \).
In Sect. 5, we show that the fractional \(\beta \)-variation weakly converges and \(\Gamma \)-converges to the fractional \(\alpha \)-variation as \(\beta \rightarrow \alpha ^-\) for any \(\alpha \in (0,1)\).
In Appendix A, for the reader’s convenience, we state and prove two known results on the truncation and the approximation of BV functions and sets with finite perimeter that are used in Sect. 3 and in Sect. 4.
2 Preliminaries
2.1 General notation
We start with a brief description of the main notation used in this paper. In order to keep the exposition the most reader-friendly as possible, we retain the same notation adopted in our previous work [27].
Given an open set \(\Omega \), we say that a set E is compactly contained in \(\Omega \), and we write \(E\Subset \Omega \), if the \({\overline{E}}\) is compact and contained in \(\Omega \). We denote by \({\mathscr {L}}^{n}\) and \({\mathscr {H}}^{\alpha }\) the n-dimensional Lebesgue measure and the \(\alpha \)-dimensional Hausdorff measure on \({\mathbb {R}}^n\) respectively, with \(\alpha \ge 0\). Unless otherwise stated, a measurable set is a \({\mathscr {L}}^{n}\)-measurable set. We also use the notation \(|E|={\mathscr {L}}^{n}(E)\). All functions we consider in this paper are Lebesgue measurable, unless otherwise stated. We denote by \(B_r(x)\) the standard open Euclidean ball with center \(x\in {\mathbb {R}}^n\) and radius \(r>0\). We let \(B_r=B_r(0)\). Recall that \(\omega _{n} := |B_1|=\pi ^{\frac{n}{2}}/\Gamma \left( \frac{n+2}{2}\right) \) and \({\mathscr {H}}^{n-1}(\partial B_{1}) = n \omega _n\), where \(\Gamma \) is Euler’s Gamma function, see [9].
We let \(\mathrm {GL}(n)\supset \mathrm {O}(n)\supset \mathrm {SO}(n)\) be the general linear group, the orthogonal group and the special orthogonal group respectively. We tacitly identify \(\mathrm {GL}(n)\subset {\mathbb {R}}^{n^2}\) with the space of invertible \(n\times n\) - matrices and we endow it with the usual Euclidean distance in \({\mathbb {R}}^{n^2}\).
For \(k \in {\mathbb {N}}_{0} \cup \{+ \infty \}\) and \(m \in {\mathbb {N}}\), we denote by \(C^{k}_{c}(\Omega ; {\mathbb {R}}^{m})\) and \({{\,\mathrm{Lip}\,}}_c(\Omega ; {\mathbb {R}}^{m})\) the spaces of \(C^{k}\)-regular and, respectively, Lipschitz-regular, m-vector-valued functions defined on \({\mathbb {R}}^n\) with compact support in \(\Omega \).
For any exponent \(p\in [1,+\infty ]\), we denote by \(L^p(\Omega ;{\mathbb {R}}^m)\) the space of m-vector-valued Lebesgue p-integrable functions on \(\Omega \). For \(p\in [1,+\infty ]\), we say that \((f_k)_{k\in {\mathbb {N}}}\subset L^p(\Omega ;{\mathbb {R}}^m)\) weakly converges to \(f\in L^p(\Omega ;{\mathbb {R}}^m)\), and we write \(f_k\rightharpoonup f\) in \(L^p(\Omega ;{\mathbb {R}}^m)\) as \(k\rightarrow +\infty \), if
for all \(\varphi \in L^q(\Omega ;{\mathbb {R}}^m)\), with \(q\in [1,+\infty ]\) the conjugate exponent of p, that is, \(\frac{1}{p}+\frac{1}{q}=1\) (with the usual convention \(\frac{1}{+\infty }=0\)). Note that in the case \(p=+\infty \) we make a little abuse of terminology, since the limit in (2.1) actually defines the weak*-convergence in \(L^\infty (\Omega ;{\mathbb {R}}^m)\).
We let
be the space of m-vector-valued Sobolev functions on \(\Omega \), see for instance [46, Chapter 10] for its precise definition and main properties. We also let
We let
be the space of m-vector-valued functions of bounded variation on \(\Omega \), see for instance [4, Chapter 3] or [34, Chapter 5] for its precise definition and main properties. We also let
For \(\alpha \in (0,1)\) and \(p\in [1,+\infty )\), we let
be the space of m-vector-valued fractional Sobolev functions on \(\Omega \), see [32] for its precise definition and main properties. We also let
For \(\alpha \in (0,1)\) and \(p=+\infty \), we simply let
so that \(W^{\alpha ,\infty }(\Omega ;{\mathbb {R}}^m)=C^{0,\alpha }_b(\Omega ;{\mathbb {R}}^m)\), the space of m-vector-valued bounded \(\alpha \)-Hölder continuous functions on \(\Omega \).
We let \({\mathscr {M}}(\Omega ;{\mathbb {R}}^m)\) be the space of m-vector-valued Radon measures with finite total variation, precisely
for \(\mu \in {\mathscr {M}}(\Omega ;{\mathbb {R}}^m)\). We say that \((\mu _k)_{k\in {\mathbb {N}}}\subset {\mathscr {M}}(\Omega ;{\mathbb {R}}^m)\) weakly converges to \(\mu \in {\mathscr {M}}(\Omega ;{\mathbb {R}}^m)\), and we write \(\mu _k\rightharpoonup \mu \) in \({\mathscr {M}}(\Omega ;{\mathbb {R}}^m)\) as \(k\rightarrow +\infty \), if
for all \(\varphi \in C_c^0(\Omega ;{\mathbb {R}}^m)\). Note that we make a little abuse of terminology, since the limit in (2.2) actually defines the weak*-convergence in \({\mathscr {M}}(\Omega ;{\mathbb {R}}^m)\).
In order to avoid heavy notation, if the elements of a function space \(F(\Omega ;{\mathbb {R}}^m)\) are real-valued (i.e. \(m=1\)), then we will drop the target space and simply write \(F(\Omega )\).
2.2 Basic properties of \(\nabla ^\alpha \) and \(\mathrm {div}^\alpha \)
We recall the non-local operators \(\nabla ^\alpha \) and \({{\,\mathrm{div}\,}}^\alpha \) introduced by Šilhavý in [72] (see also our previous work [27]).
Let \(\alpha \in (0,1)\) and set
We let
be the fractional \(\alpha \)-gradient of \(f\in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\) at \(x\in {\mathbb {R}}^n\). We also let
be the fractional \(\alpha \)-divergence of \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\) at \(x\in {\mathbb {R}}^n\). The non-local operators \(\nabla ^\alpha \) and \({{\,\mathrm{div}\,}}^\alpha \) are well defined in the sense that the involved integrals converge and the limits exist, see [72, Section 7] and [27, Section 2]. Moreover, since
it is immediate to check that \(\nabla ^{\alpha }c=0\) for all \(c\in {\mathbb {R}}\) and
for all \(f\in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\). Analogously, we also have
for all \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\).
Given \(\alpha \in (0,n)\), we let
be the Riesz potential of order \(\alpha \in (0,n)\) of a function \(u\in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^m)\). We recall that, if \(\alpha ,\beta \in (0,n)\) satisfy \(\alpha +\beta <n\), then we have the following semigroup property
for all \(u\in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^m)\). In addition, if \(1<p<q<+\infty \) satisfy
then there exists a constant \(C_{n,\alpha ,p}>0\) such that the operator in (2.3) satisfies
for all \(u\in C^\infty _c({\mathbb {R}}^n;\,{\mathbb {R}}^m)\). As a consequence, the operator in (2.3) extends to a linear continuous operator from \(L^p({\mathbb {R}}^n;{\mathbb {R}}^m)\) to \(L^q({\mathbb {R}}^n;{\mathbb {R}}^m)\), for which we retain the same notation. For a proof of (2.4) and (2.5), we refer the reader to [76, Chapter V, Section 1] and to [40, Section 1.2.1].
We can now recall the following result, see [27, Proposition 2.2 and Corollary 2.3].
Proposition 2.1
Let \(\alpha \in (0,1)\). If \(f\in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^{n})\), then
and \(\nabla ^{\alpha }f \in L^1({\mathbb {R}}^n; {\mathbb {R}}^{n})\cap L^{\infty }({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\), with
and
for any bounded open set \(U\subset {\mathbb {R}}^n\) such that \({{\,\mathrm{supp}\,}}(f) \subset U\), where
Analogously, if \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) then
and \(\mathrm {div}^{\alpha } \varphi \in L^1({\mathbb {R}}^n)\cap L^{\infty }({\mathbb {R}}^{n})\), with
and
for any bounded open set \(U\subset {\mathbb {R}}^n\) such that \({{\,\mathrm{supp}\,}}(\varphi ) \subset U\), where \(C_{n, \alpha , U}\) is as in (2.9).
2.3 Extension of \(\nabla ^\alpha \) and \(\mathrm {div}^\alpha \) to \({{\,\mathrm{Lip}\,}}_b\)-regular tests
In the following result, we extend the fractional \(\alpha \)-divergence to \({{\,\mathrm{Lip}\,}}_b\)-regular vector fields.
Lemma 2.2
(Extension of \(\mathrm {div}^\alpha \) to \({{\,\mathrm{Lip}\,}}_b\)). Let \(\alpha \in (0,1)\). The operator
given by
for all \(\varphi \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\), is well defined, with
and satisfies
for all \(x\in {\mathbb {R}}^n\). Moreover, if in addition \(I_{1-\alpha }|\mathrm {div}\varphi |\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\), then
for a.e. \(x\in {\mathbb {R}}^n\).
Proof
We split the proof in two steps.
Step 1: proof of (2.13), (2.14) and (2.15). Given \(x\in {\mathbb {R}}^n\) and \(r>0\), we can estimate
and
Hence the function in (2.13) is well defined for all \(x\in {\mathbb {R}}^n\) and
so that (2.14) follows by optimising the right-hand side in \(r>0\). Moreover, since
and
for all \(\varepsilon >0\), by Lebesgue’s Dominated Convergence Theorem we immediately get the two equalities in (2.15) for all \(x\in {\mathbb {R}}^n\).
Step 2: proof of (2.16). Assume that \(I_{1-\alpha }|\mathrm {div}\varphi |\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\). Then
for a.e. \(x\in {\mathbb {R}}^n\). Hence, by Lebesgue’s Dominated Convergence Theorem, we can write
for a.e. \(x\in {\mathbb {R}}^n\). Now let \(\varepsilon >0\) be fixed and let \(R>0\). Again by (2.17) and Lebesgue’s Dominated Convergence Theorem, we have
for a.e. \(x\in {\mathbb {R}}^n\). Moreover, integrating by parts, we get
for all \(R>0\) and for a.e. \(x\in {\mathbb {R}}^n\). Since \(\varphi \in L^\infty ({\mathbb {R}}^n;{\mathbb {R}}^n)\), by Lebesgue’s Dominated Convergence Theorem we have
for all \(\varepsilon >0\) and all \(x\in {\mathbb {R}}^n\). We can also estimate
for all \(R>0\) and all \(x\in {\mathbb {R}}^n\). We thus have that
for all \(\varepsilon >0\) and a.e. \(x\in {\mathbb {R}}^n\). Since also
for all \(\varepsilon >0\) and \(x\in {\mathbb {R}}^n\), we conclude that
for a.e. \(x\in {\mathbb {R}}^n\), proving (2.16). \(\square \)
We can also extend the fractional \(\alpha \)-gradient to \({{\,\mathrm{Lip}\,}}_b\)-regular functions. The proof is very similar to the one of Lemma 2.2 and is left to the reader.
Lemma 2.3
(Extension of \(\nabla ^\alpha \) to \({{\,\mathrm{Lip}\,}}_b\)). Let \(\alpha \in (0,1)\). The operator
given by
for all \(f\in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n)\), is well defined, with
and satisfies
for all \(x\in {\mathbb {R}}^n\). Moreover, if in addition \(I_{1-\alpha }|\nabla f|\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\), then
for a.e. \(x\in {\mathbb {R}}^n\).
2.4 Extended Leibniz’s rules for \(\nabla ^\alpha \) and \(\mathrm {div}^\alpha \)
The following two results extend the validity of Leibniz’s rules proved in [27, Lemmas 2.6 and 2.7] to \({{\,\mathrm{Lip}\,}}_b\)-regular functions and \({{\,\mathrm{Lip}\,}}_b\)-regular vector fields. The proofs are very similar to the ones given in [27] and to that of Lemma 2.2, and thus are left to the reader.
Lemma 2.4
(Extended Leibniz’s rule for \(\nabla ^\alpha \)). Let \(\alpha \in (0,1)\). If \(f\in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n)\) and \(\eta \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\), then
where
for all \(x\in {\mathbb {R}}^n\), with
and
Lemma 2.5
(Extended Leibniz’s rule for \(\mathrm {div}^\alpha \)). Let \(\alpha \in (0,1)\). If \(\varphi \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\) and \(\eta \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n)\), then
where
for all \(x\in {\mathbb {R}}^n\), with
and
2.5 Extended integration-by-part formulas
We now recall the definition of the space of functions with bounded fractional \(\alpha \)-variation. Given \(\alpha \in (0, 1)\), we let
where
is the fractional \(\alpha \)-variation of \(f\in L^1({\mathbb {R}}^n)\). We refer the reader to [27, Section 3] for the basic properties of this function space. Here we just recall the following result, see [27, Theorem 3.2 and Proposition 3.6] for the proof.
Theorem 2.6
(Structure theorem for \(BV^\alpha \) functions). Let \(\alpha \in (0,1)\). If \(f \in L^{1}({\mathbb {R}}^{n})\), then \(f \in BV^{\alpha }({\mathbb {R}}^{n})\) if and only if there exists a finite vector-valued Radon measure \(D^{\alpha } f \in {\mathscr {M}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) such that
for all \(\varphi \in {{\,\mathrm{Lip}\,}}_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\).
Thanks to Lemma 2.5, we can actually prove that a function in \(BV^\alpha ({\mathbb {R}}^n)\) can be tested against any \({{\,\mathrm{Lip}\,}}_b\)-regular vector field.
Proposition 2.7
(\({{\,\mathrm{Lip}\,}}_b\)-regular test for \(BV^\alpha \) functions). Let \(\alpha \in (0,1)\). If \(f\in BV^\alpha ({\mathbb {R}}^n)\), then (2.18) holds for all \(\varphi \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\).
Proof
We argue as in the proof of [27, Theorem 3.8]. Fix \(\varphi \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\) and let \((\eta _R)_{R>0}\subset C^\infty _c({\mathbb {R}}^n)\) be a family of cut-off functions as in [27, Section 3.3]. On the one hand, since
for all \(R>0\), by Lebesgue’s Dominated Convergence Theorem we have
On the other hand, by Lemma 2.5 we can write
for all \(R>0\). By [27, Proposition 3.6], we have
for all \(R>0\). Since
for all \(R>0\), by Lebesgue’s Dominated Convergence Theorem (with respect to the finite measure \(|D^\alpha f|\)) we have
Finally, we can estimate
and, similarly,
By Lebesgue’s Dominated Convergence Theorem, we thus get that
and the conclusion follows. \(\square \)
Thanks to Lemma 2.4, we can prove that a function in \({{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n)\) can be tested against any \({{\,\mathrm{Lip}\,}}_c\)-regular vector field. The proof is very similar to the one of Proposition 2.7 and is thus left to the reader.
Proposition 2.8
(Integration by parts for \({{\,\mathrm{Lip}\,}}_b\)-regular functions). Let \(\alpha \in (0,1)\). If \(f\in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n)\), then
for all \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\).
2.6 Comparison between \(W^{\alpha ,1}\) and \(BV^{\alpha }\) seminorms
In this section, we completely answer a question left open in [27, Section 1.4]. Given \(\alpha \in (0, 1)\) and an open set \(\Omega \subset {\mathbb {R}}^n\), we want to study the equality cases in the inequalities
as long as \(f\in W^{\alpha ,1}({\mathbb {R}}^n)\) and \(P_\alpha (E;\Omega )<+\infty \). The key idea to the solution of this problem lies in the following simple result.
Lemma 2.9
Let \(A\subset {\mathbb {R}}^n\) be a measurable set with \({\mathscr {L}}^{n}(A)>0\). If \(F \in L^{1}(A; {\mathbb {R}}^m)\), then
with equality if and only if \(F = f \nu \) a.e. in A for some constant direction \(\nu \in {\mathbb {S}}^{m - 1}\) and some scalar function \(f \in L^{1}(A)\) with \(f \ge 0\) a.e. in A.
Proof
The inequality is well known and it is obvious that it is an equality if \(F = f \nu \) a.e. in A for some constant direction \(\nu \in {\mathbb {S}}^{m-1}\) and some scalar function \(f \in L^{1}(A)\) with \(f \ge 0\) a.e. in A. So let us assume that
If \(\int _A F(x) \, dx=0\), then also \(\int _A | F(x) | \, dx=0\). Thus \(F=0\) a.e. in A and there is nothing to prove. If \(\int _A F(x) \, dx\ne 0\) instead, then we can write
with
Therefore, we obtain \(|F(x)| = F(x) \cdot \nu \) for a.e. \(x \in A\), so that \(\frac{F(x)}{|F(x)|} \cdot \nu = 1\) for a.e. \(x \in A\) such that \(|F(x)|\ne 0\). This implies that \(F = f \nu \) a.e. in A with \(f = |F|\in L^{1}(A)\) and the conclusion follows. \(\square \)
As an immediate consequence of Lemma 2.9, we have the following result.
Corollary 2.10
Let \(\alpha \in (0, 1)\). If \(f \in W^{\alpha , 1}({\mathbb {R}}^{n})\), then
with equality if and only if \(f=0\) a.e. in \({\mathbb {R}}^n\).
Proof
Inequality (2.19) was proved in [27, Theorem 3.18]. Note that, given \(f \in L^{1}({\mathbb {R}}^{n})\), \([f]_{W^{\alpha ,1}({\mathbb {R}}^n)}=0\) if and only if \(f=0\) a.e. and thus, in this case, (2.19) is trivially an equality. If (2.19) holds as an equality and f is not equivalent to the zero function, then
and thus
for all \(x\in U\), for some measurable set \(U\subset {\mathbb {R}}^n\) such that \({\mathscr {L}}^{n}({\mathbb {R}}^n\setminus U)=0\). Now let \(x\in U\) be fixed. By Lemma 2.9 (applied with \(A={\mathbb {R}}^n\)), (2.20) implies that the (non-identically zero) vector field
has constant direction for all \(y\in V_x\), for some measurable set \(V_x\subset {\mathbb {R}}^n\) such that \({\mathscr {L}}^{n}({\mathbb {R}}^n\setminus V_x)=0\). Thus, given \(y,y'\in V_x\), the two vectors \(y-x\) and \(y'-x\) are linearly dependent, so that the three points x, y and \(y'\) are collinear. If \(n\ge 2\), then this immediately gives \({\mathscr {L}}^{n}(V_x)=0\), a contradiction, so that (2.19) must be strict. If instead \(n=1\), then we know that
We claim that (2.21) implies that the function f is (equivalent to) a (non-constant) monotone function. If so, then \(f\notin L^1({\mathbb {R}})\), in contrast with the fact that \(f\in W^{\alpha ,1}({\mathbb {R}})\), so that (2.19) must be strict and the proof is concluded. To prove the claim, we argue as follows. Fix \(x\in U\) and assume that
for all \(y\in V_x\) without loss of generality. Now pick \(x'\in U\cap V_x\) such that \(x'>x\). Then, choosing \(y=x'\) in (2.22), we get \((f(x') - f(x))\,(x' - x)>0\) and thus \(f(x')>f(x)\). Similarly, if \(x'\in U\cap V_x\) is such that \(x'<x\), then \(f(x')<f(x)\). Hence
for all \(x\in U\) (where \({{\,\mathrm{ess\,sup}\,}}\) and \({{\,\mathrm{ess\,inf}\,}}\) refer to the essential supremum and the essential infimum respectively) and thus f must be equivalent to a (non-constant) non-decreasing function. \(\square \)
Given an open set \(\Omega \subset {\mathbb {R}}^n\) and a measurable set \(E\subset {\mathbb {R}}^n\), we define
It is obvious to see that
where \(P_\alpha \) is the fractional perimeter introduced in (1.10). Arguing as in the proof of [27, Proposition 4.8]it is immediate to see that
an inequality stronger than that in (1.9). In analogy with Corollary 2.10, we have the following result.
Corollary 2.11
Let \(\alpha \in (0,1)\), \(\Omega \subset {\mathbb {R}}^n\) be an open set and \(E\subset {\mathbb {R}}^n\) be a measurable set such that \(\tilde{P}_{\alpha }(E;\Omega )<+\infty \).
-
(i)
If \(n\ge 2\), \({\mathscr {L}}^{n}(E)>0\) and \({\mathscr {L}}^{n}({\mathbb {R}}^n\setminus E)>0\), then inequality (2.23) is strict.
-
(ii)
If \(n=1\), then (2.23) is an equality if and only if the following hold:
-
(a)
for a.e. \(x\in \Omega \cap E\), \({\mathscr {L}}^{1}((-\infty ,x)\setminus E)=0\) vel \({\mathscr {L}}^{1}((x,+\infty )\setminus E)=0\);
-
(b)
for a.e. \(x\in \Omega \setminus E\), \({\mathscr {L}}^{1}((-\infty ,x) \cap E)=0\) vel \({\mathscr {L}}^{1}((x,+\infty ) \cap E)=0\).
-
(a)
Proof
We prove the two statements separately.
Proof of (i). Assume \(n\ge 2\). Since \({\mathscr {L}}^{n}(E)>0\), for a given \(x\in \Omega \setminus E\) the map
does not have constant orientation. Similarly, since \({\mathscr {L}}^{n}({\mathbb {R}}^n\setminus E)>0\), for a given \(x\in \Omega \cap E\) also the map
does not have constant orientation. Hence, by Lemma 2.9, we must have
and, similarly,
We thus get
proving (i).
Proof of (ii). Assume \(n=1\). We argue as in the proof of [27, Proposition 4.12]. Let
Then we can write
and
Hence (2.23) is an equality if and only if
for a.e. \(x\in \Omega \). Observing that
for a.e. \(x\in \Omega \), we deduce that (2.23) is an equality if and only if
for a.e. \(x\in \Omega \). Now, on the one hand, squaring both sides of (2.25) and simplifying, we get that (2.23) is an equality if and only if
for a.e. \(x\in \Omega \). On the other hand, we can rewrite (2.26) as
for a.e. \(x\in \Omega \), so that we must have
and
for a.e. \(x\in \Omega \). Hence (2.27) can be equivalently rewritten as
for a.e. \(x\in \Omega \). Thus (2.23) is an equality if and only if at least one of the two integrals in the left-hand side of (2.28) is zero, and the reader can check that (ii) readily follows. \(\square \)
Remark 2.12
(Half-lines in Corollary 2.11(ii)) In the case \(n=1\), it is worth to stress that (2.23) is always an equality when the set \(E\subset {\mathbb {R}}\) is (equivalent to) an half-line, i.e.,
for any \(\alpha \in (0,1)\), any \(a\in {\mathbb {R}}\) and any open set \(\Omega \subset {\mathbb {R}}\) such that \(\tilde{P}_{\alpha }((a,+\infty );\Omega )<+\infty \). However, the equality cases in (2.23) are considerably richer. Indeed, on the one side,
and, on the other side,
for any \(\alpha \in (0,1)\). We leave the simple computations to the interested reader.
3 Estimates and representation formulas for the fractional \(\alpha \)-gradient
3.1 Integrability properties of the fractional \(\alpha \)-gradient
We begin with the following technical local estimate on the \(W^{\alpha ,1}\)-seminorm of a function in \(BV_{{{\,\mathrm{loc}\,}}}\).
Lemma 3.1
Let \(\alpha \in (0,1)\) and let \(f\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\). Then \(f\in W^{\alpha ,1}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) with
for all \(R>0\).
Proof
Fix \(R>0\) and let \(f\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) be such that \(f\in C^1(B_{3R})\). We can estimate
Since
for all \(h\in {\mathbb {R}}^n\), we have
proving (3.1) for all \(f\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\cap C^1(B_{3R})\). Now fix \(R>0\) and let \(f\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\). By [34, Theorem 5.3], there exists \((f_k)_{k\in {\mathbb {N}}}\subset BV(B_{3R})\cap C^\infty (B_{3R})\) such that \(|Df_k|(B_{3R})\rightarrow |Df|(B_{3R})\) and \(f_k\rightarrow f\) a.e. in \(B_{3R}\) as \(k\rightarrow +\infty \). The conclusion thus follows by a simple application of Fatou’s Lemma. \(\square \)
In the following result, we collect several local integrability estimates involving the fractional \(\alpha \)-gradient of a function satisfying various regularity assumptions.
Proposition 3.2
The following statements hold.
-
(i)
If \(f\in BV({\mathbb {R}}^n)\), then \(f\in BV^\alpha ({\mathbb {R}}^n)\) for all \(\alpha \in (0,1)\) with \(D^\alpha f=\nabla ^\alpha f{\mathscr {L}}^{n}\) and
$$\begin{aligned} \nabla ^{\alpha } f = I_{1-\alpha }D f \quad \text {a.e. in } {\mathbb {R}}^{n}. \end{aligned}$$(3.2)In addition, for any bounded open set \(U\subset {\mathbb {R}}^n\), we have
$$\begin{aligned} \Vert \nabla ^{\alpha } f\Vert _{L^{1}(U;\, {\mathbb {R}}^{n})} \le C_{n, \alpha , U} \, |D f|({\mathbb {R}}^{n}) \end{aligned}$$(3.3)for all \(\alpha \in (0,1)\), where \(C_{n, \alpha , U}\) is as in (2.9). Finally, given an open set \(A\subset {\mathbb {R}}^n\), we have
$$\begin{aligned}&\Vert \nabla ^\alpha f\Vert _{L^1(A;\,{\mathbb {R}}^n)}\nonumber \\&\quad \le \frac{n\omega _n\,\mu _{n,\alpha }}{n+\alpha -1}\left( \frac{|Df|( {}\overline{A_r})}{1-\alpha }\,r^{1-\alpha }+\frac{n+2\alpha -1}{\alpha }\,\Vert f\Vert _{L^1({\mathbb {R}}^n)}\,r^{-\alpha } \right) \end{aligned}$$(3.4)for all \(r>0\) and \(\alpha \in (0,1)\), where \(A_r:=\Bigg \{x\in {\mathbb {R}}^n : {{\,\mathrm{dist}\,}}(x,A)<r\Bigg \}\). In particular, we have
$$\begin{aligned} \Vert \nabla ^\alpha f\Vert _{L^1({\mathbb {R}}^n;\,{\mathbb {R}}^n)} \le \frac{n\omega _n\,\mu _{n,\alpha } (n + 2 \alpha - 1)^{1 - \alpha }}{\alpha (1 - \alpha )(n+\alpha -1)}\, \Vert f\Vert _{L^{1}({\mathbb {R}}^{n})}^{1 - \alpha }\, [f]_{BV({\mathbb {R}}^n)}^{\alpha }. \end{aligned}$$(3.5) -
(ii)
If \(f \in L^{\infty }({\mathbb {R}}^{n})\cap W^{\alpha , 1}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\), then the weak fractional \(\alpha \)-gradient \(D^{\alpha } f \in {\mathscr {M}}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n; {\mathbb {R}}^n)\) exists and satisfies \(D^\alpha f=\nabla ^\alpha f{\mathscr {L}}^{n}\) with \(\nabla ^{\alpha } f \in L^{1}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) and
$$\begin{aligned} \begin{aligned} \Vert \nabla ^{\alpha } f \Vert _{L^{1}(B_{R};\, {\mathbb {R}}^{n})}&\le \mu _{n, \alpha } \int _{B_{R}} \int _{{\mathbb {R}}^{n}} \frac{|f(x) - f(y)|}{|x - y|^{n + \alpha }} \, dx \, dy\\&\le \mu _{n, \alpha } \left( [f]_{W^{\alpha , 1}(B_{R})} + P_{\alpha }(B_{R})\, \Vert f\Vert _{L^{\infty }({\mathbb {R}}^{n})} \right) \end{aligned} \end{aligned}$$(3.6)for all \(R>0\) and \(\alpha \in (0,1)\).
-
(iii)
If \(f \in L^{\infty }({\mathbb {R}}^{n})\cap BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\), then the weak fractional \(\alpha \)-gradient \(D^{\alpha } f \in {\mathscr {M}}_\mathrm{loc}({\mathbb {R}}^n; {\mathbb {R}}^n)\) exists and satisfies \(D^\alpha f=\nabla ^\alpha f{\mathscr {L}}^{n}\) with \(\nabla ^{\alpha } f \in L^{1}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) and
$$\begin{aligned}&\Vert \nabla ^{\alpha } f \Vert _{L^{1}(B_{R};\, {\mathbb {R}}^{n})}\nonumber \\&\quad \le \mu _{n, \alpha }\left( \frac{n\omega _n(2R)^{1-\alpha }}{1-\alpha } \, |Df|(B_{3R}) + \frac{2^{\alpha + 1} (n \omega _n)^2 R^{n-\alpha }}{\alpha \,\Gamma (1-\alpha )^{-1}}\, \Vert f\Vert _{L^{\infty }({\mathbb {R}}^{n})}\right) .\qquad \qquad \end{aligned}$$(3.7)for all \(R>0\) and \(\alpha \in (0,1)\).
Proof
We prove the three statements separately.
Proof of (i). Thanks to [27, Theorem 3.18], we just need to prove (3.3) and (3.4).
We prove (3.3). By (3.2), by Tonelli’s Theorem and by [27, Lemma 2.4], we get
where \(C_{n, \alpha , U}\) is defined as in (2.9).
We now prove (3.4) in two steps.
Proof of (3.4), Step 1. Assume \(f\in C^\infty _c({\mathbb {R}}^n)\) and fix \(r>0\). We have
We estimate the two double integrals appearing in the right-hand side separately. By Tonelli’s Theorem, we have
Concerning the second double integral, integrating by parts we get
for all \(x\in A\). Hence, we can estimate
Thus (3.4) follows for all \(f\in C^\infty _c({\mathbb {R}}^n)\) and \(r>0\).
Proof of (3.4), Step 2. Let \(f\in BV({\mathbb {R}}^n)\) and fix \(r>0\). Combining [34, Theorem 5.3] with a standard cut-off approximation argument, we find \((f_k)_{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^n)\) such that \(f_k\rightarrow f\) in \(L^1({\mathbb {R}}^n)\) and \(|Df_k|({\mathbb {R}}^n)\rightarrow |Df|({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \). By Step 1, we have that
for all \(k\in {\mathbb {N}}\). We claim that
Indeed, if \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\), then \(\mathrm {div}^\alpha \varphi \in L^\infty ({\mathbb {R}}^n)\) by (2.12) and thus
for all \(k\in {\mathbb {N}}\), so that
Now fix \(\varphi \in C_c^0({\mathbb {R}}^n;{\mathbb {R}}^n)\). Let \(U\subset {\mathbb {R}}^n\) be a bounded open set such that \({{\,\mathrm{supp}\,}}\varphi \subset U\). For each \(\varepsilon >0\) sufficiently small, pick \(\psi _\varepsilon \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\) such that \(\Vert \varphi -\psi _\varepsilon \Vert _{L^\infty ({\mathbb {R}}^n;\,{\mathbb {R}}^n)}<\varepsilon \) and \({{\,\mathrm{supp}\,}}\psi _\varepsilon \subset U\). Then
so that
Thus, (3.9) follows passing to the limit as \(\varepsilon \rightarrow 0^+\). Thanks to (3.9), by [50, Proposition 4.29]we get that
Since
for any open set \(U\subset {\mathbb {R}}^n\) by [34, Theorem 5.2], we can estimate
Thus, (3.4) follows taking limits as \(k\rightarrow +\infty \) in (3.8). Finally, (3.5) is easily deduced by optimising the right-hand side of (3.4) in the case \(A={\mathbb {R}}^n\) with respect to \(r > 0\).
Proof of (ii). Assume \(f \in L^{\infty }({\mathbb {R}}^{n})\cap W^{\alpha , 1}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\). Given \(R>0\), we can estimate
and (3.6) follows. To prove that \(D^\alpha f=\nabla ^\alpha f{\mathscr {L}}^{n}\), we argue as in the proof of [27, Proposition 4.8]. Let \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\). Since \(f\in L^\infty ({\mathbb {R}}^n)\), we have
Hence, by the definition of \(\mathrm {div}^\alpha \) on \({{\,\mathrm{Lip}\,}}_c\)-regular vector fields (see [27, Section 2.2]) and by Lebesgue’s Dominated Convergence Theorem, we have
Since
for all \(\varepsilon >0\), by Fubini’s Theorem we can compute
Since
for all \(y\in {\mathbb {R}}^n\) and \(\varepsilon >0\), and
by (3.6), again by Lebesgue’s Dominated Convergence Theorem we conclude that
for all \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\). Thus \(D^\alpha f\in {\mathscr {M}}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n;{\mathbb {R}}^n)\) is well defined and \(D^\alpha f=\nabla ^\alpha f{\mathscr {L}}^{n}\).
Proof of (iii). Assume \(f\in L^{\infty }({\mathbb {R}}^{n})\cap BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\). By Lemma 3.1, we know that \(f \in L^{\infty }({\mathbb {R}}^{n})\cap W^{\alpha , 1}_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\) for all \(\alpha \in (0,1)\), so that \(D^{\alpha } f \in {\mathscr {M}}_\mathrm{loc}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) exists by (ii). Hence, inserting (3.1) in (3.6), we find
Since for all \(x\in B_1\) we have
being \(\Gamma \) log-convex on \((0,+\infty )\) (see [9]), we can estimate
so that
proving (3.7). \(\square \)
Note that Proposition 3.2(i), in particular, applies to any \(f\in W^{1,1}({\mathbb {R}}^n)\). In the following result, we prove that a similar result holds also for any \(f\in W^{1,p}({\mathbb {R}}^n)\) with \(p\in (1,+\infty )\).
Proposition 3.3
(\(W^{1,p}({\mathbb {R}}^n)\subset S^{\alpha ,p}({\mathbb {R}}^n)\) for \(p\in (1,+\infty )\)) Let \(\alpha \in (0,1)\) and \(p\in (1,+\infty )\). If \(f\in W^{1,p}({\mathbb {R}}^n)\), then \(f\in S^{\alpha ,p}({\mathbb {R}}^n)\) with
for any \(r>0\) and any open set \(A\subset {\mathbb {R}}^n\), where \(A_r:=\Bigg \{x\in {\mathbb {R}}^n : {{\,\mathrm{dist}\,}}(x,A)<r\Bigg \}\). In particular, we have
In addition, if \(p\in \big (1,\frac{n}{1-\alpha }\big )\) and \(q=\frac{np}{n-(1-\alpha )p}\), then
and \(\nabla ^\alpha _w f\in L^q({\mathbb {R}}^n;{\mathbb {R}}^n)\).
Proof
We argue as in the proof of Proposition 3.2(i).
Proof of (3.10). The proof of (3.10) for all \(f\in C^\infty _c({\mathbb {R}}^n)\) is very similar to that of (3.4) and is thus left to the reader. Now let \(f\in W^{1,p}({\mathbb {R}}^n)\) and fix an open set \(A\subset {\mathbb {R}}^n\) and \(r>0\). Combining [34, Theorem 4.2] with a standard cut-off approximation argument, we find \((f_k)_{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^n)\) such that \(f_k\rightarrow f\) in \(W^{1,p}({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \). We thus have that
for all \(k\in {\mathbb {N}}\). Hence, choosing \(A={\mathbb {R}}^n\), we get that the sequence \((\nabla ^\alpha f_k)_{k\in {\mathbb {N}}}\) is uniformly bounded in \(L^p({\mathbb {R}}^n;{\mathbb {R}}^n)\). Up to pass to a subsequence (which we do not relabel for simplicity), there exists \(g\in L^p({\mathbb {R}}^n;{\mathbb {R}}^n)\) such that \(\nabla ^\alpha f_k\rightharpoonup g\) in \(L^p({\mathbb {R}}^n;{\mathbb {R}}^n)\) as \(k\rightarrow +\infty \). Given \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), we have
for all \(k\in {\mathbb {N}}\). Passing to the limit as \(k\rightarrow +\infty \), by Proposition 2.1 we get that
for any \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), so that \(g=\nabla ^\alpha _w f\) and hence \(f\in S^{\alpha ,p}({\mathbb {R}}^n)\) according to [27, Definition 3.19]. We thus have that
for any open set \(A\subset {\mathbb {R}}^n\), since
for all \(\varphi \in C^\infty _c(A;{\mathbb {R}}^n)\). Therefore, (3.10) follows by taking limits as \(k\rightarrow +\infty \) in (3.13).
Proof of (3.11). Inequality (3.11) follows by applying (3.10) with \(A={\mathbb {R}}^n\) and minimising the right-hand side with respect to \(r>0\).
Proof of (3.12). Now assume \(p\in \big (1,\frac{n}{1-\alpha }\big )\) and let \(q=\frac{np}{n-(1-\alpha )p}\). Let \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\) be fixed. Recalling inequality (2.5), since \(\varphi \in L^{\frac{q}{q-1}}({\mathbb {R}}^n;{\mathbb {R}}^n)\) we have that
In particular, Fubini’s Theorem implies that
Since \(\mathrm {div}^{\alpha } \varphi \in L^{\frac{p}{p-1}}({\mathbb {R}}^n)\) by Proposition 2.1, we also get that
Therefore, observing that \(I_{1-\alpha }\varphi \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\) because \(\nabla I_{1 - \alpha } \varphi = \nabla ^{\alpha } \varphi \in L^{\infty }({\mathbb {R}}^{n}; {\mathbb {R}}^{n^2})\) again by Proposition 2.1 and performing a standard cut-off approximation argument, we can integrate by parts and obtain
Therefore
for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), proving (3.12). In particular, notice that \(\nabla ^\alpha _w f\in L^q({\mathbb {R}}^n;{\mathbb {R}}^n)\) by inequality (2.5). The proof is complete. \(\square \)
For the case \(p=+\infty \), we have the following immediate consequence of Lemma 2.4 and Proposition 2.8.
Corollary 3.4
(\(W^{1,\infty }({\mathbb {R}}^n)\subset S^{\alpha ,\infty }({\mathbb {R}}^n)\)) Let \(\alpha \in (0,1)\). If \(f\in W^{1,\infty }({\mathbb {R}}^n)\), then \(f\in S^{\alpha ,\infty }({\mathbb {R}}^n)\) with
3.2 Two representation formulas for the \(\alpha \)-variation
In this section, we prove two useful representation formulas for the \(\alpha \)-variation.
We begin with the following weak representation formula for the fractional \(\alpha \)-variation of functions in \(BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n) \cap L^\infty ({\mathbb {R}}^n)\). Here and in the following, we denote by \(f^\star \) the precise representative of \(f\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\), see (A.1) for the definition.
Proposition 3.5
Let \(\alpha \in (0, 1)\) and \(f \in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n) \cap L^\infty ({\mathbb {R}}^n)\). Then \(\nabla ^\alpha f \in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n;{\mathbb {R}}^n)\) and
for all \(\varphi \in {{\,\mathrm{Lip}\,}}_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\).
Proof
By Proposition 3.2(iii), we know that \(\nabla ^{\alpha } f \in L^{1}_\mathrm{loc}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) for all \(\alpha \in (0,1)\). By Theorem A.1, we also know that \(f\chi _{B_R}\in BV({\mathbb {R}}^n)\cap L^\infty ({\mathbb {R}}^n)\) with \(D(\chi _{B_{R}} f) = \chi _{B_{R}}^\star D f + f^\star D \chi _{B_{R}}\) for all \(R>0\). Now fix \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\) and take \(R > 0\) such that \({{\,\mathrm{supp}\,}}\varphi \subset B_{R/2}\). By [27, Theorem 3.18], we have that
Moreover, we can split the last integral as
For all \(x\in B_{R/2}\), we can estimate
and so, since \({{\,\mathrm{supp}\,}}\varphi \subset B_{R/2}\), we get that
Therefore, by (2.11), Lebesgue’s Dominated Convergence Theorem, (3.16) and (3.17), we get that
and the conclusion follows. \(\square \)
In the following result, we show that for all functions in \(bv({\mathbb {R}}^n)\cap L^\infty ({\mathbb {R}}^n)\) one can actually pass to the limit as \(R\rightarrow +\infty \) inside the integral in the right-hand side of (3.15).
Corollary 3.6
If either \(f\in BV({\mathbb {R}}^n)\) or \(f\in bv({\mathbb {R}}^n)\cap L^\infty ({\mathbb {R}}^n)\), then
Proof
If \(f\in BV({\mathbb {R}}^n)\), then (3.18) coincides with (3.2) and there is nothing to prove. So let us assume that \(f\in bv({\mathbb {R}}^n)\cap L^\infty ({\mathbb {R}}^n)\). Writing \(Df = \nu _{f} |D f|\) with \(\nu _f\in {\mathbb {S}}^{n-1}\) |Df|-a.e. in \({\mathbb {R}}^n\), for all \(x\in {\mathbb {R}}^n\) we have
Moreover, for all \(x\in {\mathbb {R}}^n\), we have
because \(I_{1-\alpha }|Df|\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) by [27, Lemma 2.4]. Therefore, by Lebesgue’s Dominated Convergence Theorem (applied with respect to the finite measure |Df|), we get that
Now let \(\varphi \in {{\,\mathrm{Lip}\,}}_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\). Since
again by Lebesgue’s Dominated Convergence Theorem we get that
The conclusion thus follows by combining (3.15) with (3.19). \(\square \)
3.3 Relation between \(BV^\beta \) and \(BV^{\alpha , p}\) for \(\beta <\alpha \) and \(p > 1\)
Let us recall the following result, see [27, Lemma 3.28].
Lemma 3.7
Let \(\alpha \in (0,1)\). The following properties hold.
-
(i)
If \(f\in BV^\alpha ({\mathbb {R}}^n)\), then \(u:=I_{1-\alpha }f\in bv({\mathbb {R}}^n)\) with \(Du=D^\alpha f\) in \({\mathscr {M}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\).
-
(ii)
If \(u\in BV({\mathbb {R}}^n)\), then \(f:= (-\Delta )^{\frac{1-\alpha }{2}}u\in BV^\alpha ({\mathbb {R}}^n)\) with
$$\begin{aligned} \Vert f\Vert _{L^1({\mathbb {R}}^n)}\le c_{n,\alpha }\Vert u\Vert _{BV({\mathbb {R}}^n)} \quad \text {and}\quad D^\alpha f=D u \quad \text {in } {\mathscr {M}}({\mathbb {R}}^{n}; {\mathbb {R}}^{n}). \end{aligned}$$As a consequence, the operator \((-\Delta )^{\frac{1-\alpha }{2}}:BV({\mathbb {R}}^n)\rightarrow BV^\alpha ({\mathbb {R}}^n)\) is continuous.
We can thus relate functions with bounded \(\alpha \)-variation and functions with bounded variation via Riesz potential and the fractional Laplacian. We would like to prove a similar result between functions with bounded \(\alpha \)-variation and functions with bounded \(\beta \)-variation, for any couple of exponents \(0<\beta<\alpha < 1\).
However, although the standard variation of a function \(f\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) is well defined, it is not clear whether the functional
is well posed for all \(\varphi \in C^{\infty }_{c}({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\), since \(\mathrm {div}^{\alpha } \varphi \) does not have compact support. Nevertheless, thanks to Proposition 2.1, the functional in (3.20) is well defined as soon as \(f\in L^p({\mathbb {R}}^n)\) for some \(p\in [1,+\infty ]\). Hence, it seems natural to define the space
for any \(\alpha \in (0, 1)\) and \(p \in [1, + \infty ]\). In particular, \(BV^{\alpha , 1}({\mathbb {R}}^{n}) = BV^{\alpha }({\mathbb {R}}^{n})\). Similarly, we let
for all \(p \in [1, + \infty ]\). In particular, \(BV^{1, 1}({\mathbb {R}}^{n}) = BV({\mathbb {R}}^{n})\).
A further justification for the definition of these new spaces comes from the following fractional version of the Gagliardo–Nirenberg–Sobolev embedding: if \(n \ge 2\) and \(\alpha \in (0, 1)\), then \(BV^{\alpha }({\mathbb {R}}^{n})\) is continuously embedded in \(L^{p}({\mathbb {R}}^{n})\) for all \(p \in \left[ 1, \frac{n}{n - \alpha } \right] \), see [27, Theorem 3.9]. Hence, thanks to (3.21), we can equivalently write
with continuous embedding for all \(n \ge 2\), \(\alpha \in (0, 1)\) and \(p \in \left[ 1, \frac{n}{n - \alpha } \right] \).
Incidentally, we remark that the continuous embedding \(BV^\alpha ({\mathbb {R}}^n)\subset L^{\frac{n}{n-\alpha }}({\mathbb {R}}^n)\) for \(n\ge 2\) and \(\alpha \in (0,1)\) can be improved using the main result of the recent work [73] (see also [74]). Indeed, if \(n\ge 2\), \(\alpha \in (0,1)\) and \(f\in C^\infty _c({\mathbb {R}}^n)\), then, by taking \(F=\nabla ^\alpha f\) in [73, Theorem 1.1], we have that
thanks to the boundedness of the Riesz transform \(R:L^{\frac{n}{n-\alpha },1}({\mathbb {R}}^n)\rightarrow L^{\frac{n}{n-\alpha },1}({\mathbb {R}}^n;{\mathbb {R}}^n)\), where \(c_{n,\alpha },c_{n,\alpha }'>0\) are two constants depending only on n and \(\alpha \), and \(L^{\frac{n}{n-\alpha },1}({\mathbb {R}}^n)\) is the Lorentz space of exponents \(\frac{n}{n-\alpha },1\) (we refer to [39, 40] for an account on Lorentz spaces and on the properties of Riesz transform). Thus, recalling [27, Theorem 3.8], we readily deduce the continuous embedding \(BV^\alpha ({\mathbb {R}}^n)\subset L^{\frac{n}{n-\alpha },1}({\mathbb {R}}^n)\) for \(n\ge 2\) and \(\alpha \in (0,1)\) by [39, Exercise 1.1.1(b)] and Fatou’s Lemma. This suggests that the spaces defined in (3.21) may be further enlarged by considering functions belonging to some Lorentz space, but we do not need this level of generality here.
In the case \(n=1\), the space \(BV^\alpha ({\mathbb {R}})\) does not embed in \(L^{\frac{1}{1-\alpha }}({\mathbb {R}})\) with continuity, see [27, Remark 3.10]. However, somehow completing the picture provided by [73], we can prove that the space \(BV^\alpha ({\mathbb {R}})\) continuously embeds in the Lorentz space \(L^{\frac{1}{1 - \alpha }, \infty }({\mathbb {R}})\). Although this result is truly interesting only for \(n=1\), we prove it below in all dimensions for the sake of completeness.
Theorem 3.8
(Weak Gagliardo–Nirenberg–Sobolev inequality). Let \(\alpha \in (0,1)\). There exists a constant \(c_{n,\alpha }>0\) such that
for all \(f\in BV^\alpha ({\mathbb {R}}^n)\). As a consequence, \(BV^{\alpha }({\mathbb {R}}^{n})\) is continuously embedded in \(L^q({\mathbb {R}}^n)\) for any \(q\in [1,\frac{n}{n-\alpha })\).
Proof
Let \(f\in C^\infty _c({\mathbb {R}}^n)\). By [72, Theorem 3.5] (see also [27, Section 3.6]), we have
so that
Since \(I_\alpha :L^1({\mathbb {R}}^n)\rightarrow L^{\frac{n}{n-\alpha },\infty }({\mathbb {R}}^n)\) is a continuous operator by Hardy–Littlewood–Sobolev inequality (see [76, Theorem 1, Chapter V] or [39, Theorem 1.2.3]), we can estimate
where \(c_{n,\alpha }>0\) is a constant depending only on n and \(\alpha \). Thus, inequality (3.22) follows for all \(f\in C^\infty _c({\mathbb {R}}^n)\). Now let \(f\in BV^\alpha ({\mathbb {R}}^n)\). By [27, Theorem 3.8], there exists \((f_k)_{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^n)\) such that \(f_k\rightarrow f\) a.e. in \({\mathbb {R}}^n\) and \(|D^\alpha f_k|({\mathbb {R}}^n)\rightarrow |D^\alpha f|({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \). By [39, Exercise 1.1.1(b)] and Fatou’s Lemma, we thus get
and so (3.22) readily follows. Finally, thanks to [39, Proposition 1.1.14], we obtain the continuous embedding of \(BV^{\alpha }({\mathbb {R}}^{n})\) in \(L^{q}({\mathbb {R}}^{n})\) for all \(q \in [1, \frac{n}{n - \alpha } )\). \(\square \)
Remark 3.9
(The embedding \(BV^{\alpha }({\mathbb {R}})\subset L^{\frac{1}{1 - \alpha }, \infty }({\mathbb {R}})\) is sharp) Let \(\alpha \in (0,1)\). The continuous embedding \(BV^{\alpha }({\mathbb {R}})\subset L^{\frac{1}{1 - \alpha }, \infty }({\mathbb {R}})\) is sharp at the level of Lorentz spaces, in the sense that \(BV^\alpha ({\mathbb {R}}^n)\setminus L^{\frac{1}{1 - \alpha }, q}({\mathbb {R}})\ne \varnothing \) for any \(q \in [1, + \infty )\). Indeed, if we let
then \(f_{\alpha } \in BV^{\alpha }({\mathbb {R}})\) by [27, Theorem 3.26], and it is not difficult to prove that \(f_{\alpha } \in L^{\frac{1}{1 - \alpha }, \infty }({\mathbb {R}})\). However, we can find a constant \(c_{\alpha } > 0\) such that
so that \(d_{f_{\alpha }} \ge d_{g_{\alpha }}\), where \(d_{f_{\alpha }}\) and \(d_{g_{\alpha }}\) are the distribution functions of \(f_{\alpha }\) and \(g_{\alpha }\). A simple calculation shows that
so that, by [39, Proposition 1.4.9], we obtain
and thus \(f_{\alpha } \notin L^{\frac{1}{1-\alpha },q}({\mathbb {R}})\) for any \(q\in [1,+\infty )\).
We collect the above continuous embeddings in the following statement.
Corollary 3.10
(The embedding \(BV^{\alpha }\subset BV^{\alpha , p}\)) Let \(\alpha \in (0, 1)\) and \(p \in \left[ 1, \frac{n}{n - \alpha }\right) \). We have \(BV^{\alpha }({\mathbb {R}}^{n}) \subset BV^{\alpha , p}({\mathbb {R}}^{n})\) with continuous embedding. In addition, if \(n \ge 2\), then also \(BV^{\alpha }({\mathbb {R}}^{n}) \subset BV^{\alpha , \frac{n}{n - \alpha }}({\mathbb {R}}^{n})\) with continuous embedding.
With Corollary 3.10 at hands, we are finally ready to investigate the relation between \(\alpha \)-variation and \(\beta \)-variation for \(0< \beta< \alpha < 1\).
Lemma 3.11
Let \(0<\beta<\alpha <1\). The following hold.
-
(i)
If \(f\in BV^\beta ({\mathbb {R}}^n)\), then \(u:=I_{\alpha -\beta }f\in BV^{\alpha , p}({\mathbb {R}}^n)\) for any \(p \in \left( \frac{n}{n - \alpha + \beta }, \frac{n}{n - \alpha } \right) \) (including \(p = \frac{n}{n - \alpha }\) if \(n \ge 2\)), with \(D^\alpha u=D^\beta f\) in \({\mathscr {M}}({\mathbb {R}}^n;{\mathbb {R}}^n)\).
-
(ii)
If \(u\in BV^\alpha ({\mathbb {R}}^n)\), then \(f:=(-\Delta )^{\frac{\alpha -\beta }{2}}u\in BV^\beta ({\mathbb {R}}^n)\) with
$$\begin{aligned} \Vert f\Vert _{L^1({\mathbb {R}}^n)}\le c_{n,\alpha ,\beta }\,\Vert u\Vert _{BV^\alpha ({\mathbb {R}}^n)} \quad \text {and}\quad D^\beta f=D^\alpha u \quad \text {in}\ {\mathscr {M}}({\mathbb {R}}^n;{\mathbb {R}}^n). \end{aligned}$$
As a consequence, the operator \((-\Delta )^{\frac{\alpha -\beta }{2}}:BV^\alpha ({\mathbb {R}}^n)\rightarrow BV^\beta ({\mathbb {R}}^n)\) is continuous.
Proof
We begin with the following observation. Let \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\) and let \(U\subset {\mathbb {R}}^n\) be a bounded open set such that \({{\,\mathrm{supp}\,}}\varphi \subset U\). By Proposition 2.1 and the semigroup property (2.4) of the Riesz potential, we can write
Similarly, we also have
so that \(I_{\alpha -\beta }|\mathrm {div}^\alpha \varphi |\in L^\infty ({\mathbb {R}}^n)\) with
by [27, Lemma 2.4] We now prove the two statements separately.
Proof of (i). Let \(f\in BV^\beta ({\mathbb {R}}^n)\) and \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\). Thanks to Corollary 3.10, if \(n \ge 2\), then \(f \in BV^{\beta , q}({\mathbb {R}}^{n})\) for any \(q \in [1, \frac{n}{n - \beta }]\) and so \(I_{\alpha - \beta } f \in L^{p}({\mathbb {R}}^{n})\) for any \(p \in \left( \frac{n}{n - \alpha + \beta }, \frac{n}{n - \alpha } \right] \) by (2.5). If instead \(n = 1\), then \(f \in BV^{\beta , q}({\mathbb {R}})\) for any \(q \in [1, \frac{1}{1 - \beta })\) and so \(I_{\alpha - \beta } f \in L^{p}({\mathbb {R}})\) for any \(p \in \left( \frac{1}{1 - \alpha + \beta }, \frac{1}{1 - \alpha } \right) \). Since \(f\in L^1({\mathbb {R}}^n)\) and \(I_{\alpha -\beta }|\mathrm {div}^\alpha \varphi |\in L^\infty ({\mathbb {R}}^n)\), by Fubini’s Theorem we have
proving that \(u:=I_{\alpha -\beta }f\in BV^{\alpha , p}({\mathbb {R}}^n)\) for any \(p \in \left( \frac{n}{n - \alpha + \beta }, \frac{n}{n - \alpha } \right) \) (including \(p = \frac{n}{n - \alpha }\) if \(n \ge 2\)), with \(D^\alpha u=D^\beta f\) in \({\mathscr {M}}({\mathbb {R}}^n;{\mathbb {R}}^n)\).
Proof of (ii). Let \(u\in BV^\alpha ({\mathbb {R}}^n)\). By [27, Theorem 3.32] we know that \(u\in W^{\alpha -\beta ,1}({\mathbb {R}}^n)\), so that \(f:=(-\Delta )^{\frac{\alpha -\beta }{2}}u\in L^1({\mathbb {R}}^n)\) with \(\Vert f\Vert _{L^1({\mathbb {R}}^n)}\le c_{n,\alpha ,\beta }\,\Vert u\Vert _{BV^\alpha ({\mathbb {R}}^n)}\), see [27, Section 3.10] Then, arguing as before, for any \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\) we get (3.23), since we have \(I_{\alpha -\beta }f =u\) in \(L^1({\mathbb {R}}^n)\) (see [27, Section 3.10]). The proof is complete. \(\square \)
3.4 The inclusion \(BV^\alpha \subset W^{\beta ,1}\) for \(\beta <\alpha \): a representation formula
In [27, Theorem 3.32], we proved that the inclusion \(BV^\alpha \subset W^{\beta ,1}\) is continuous for \(\beta <\alpha \). In the following result we prove a useful representation formula for the fractional \(\beta \)-gradient of any \(f\in BV^\alpha ({\mathbb {R}}^n)\), extending the formula obtained in Corollary 3.6.
Proposition 3.12
Let \(\alpha \in (0,1)\). If \(f\in BV^\alpha ({\mathbb {R}}^n)\), then \(f\in W^{\beta ,1}({\mathbb {R}}^n)\) for all \(\beta \in (0,\alpha )\) with
In addition, for any bounded open set \(U\subset {\mathbb {R}}^n\), we have
for all \(\beta \in (0,\alpha )\), where \(C_{n,\alpha ,U}\) is as in (2.9). Finally, given an open set \(A\subset {\mathbb {R}}^n\), we have
for all \(r>0\) and all \(\beta \in (0,\alpha )\), where \(\omega _{n,\alpha }:=\Vert \nabla ^\alpha \chi _{B_1}\Vert _{L^{1}({\mathbb {R}}^n; {\mathbb {R}}^{n})}\), \(\omega _{n, 1} := |D \chi _{B_{1}}|({\mathbb {R}}^{n}) = n \omega _{n}\), and, as above, \(A_r:=\Bigg \{x\in {\mathbb {R}}^n : {{\,\mathrm{dist}\,}}(x,A)<r\Bigg \}\). In particular, we have
Proof
Fix \(\beta \in (0,\alpha )\). By [27, Theorem 3.32] we already know that \(f\in W^{\beta ,1}({\mathbb {R}}^n)\), with \(D^\beta f=\nabla ^\beta f{\mathscr {L}}^{n}\) according to [27, Theorem 3.18]. We thus just need to prove (3.24), (3.25) and (3.26).
We prove (3.24). Let \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\). Note that \(I_{\alpha -\beta }\varphi \in {{\,\mathrm{Lip}\,}}_b({\mathbb {R}}^n;{\mathbb {R}}^n)\) is such that \(\mathrm {div}I_{\alpha -\beta }\varphi =I_{\alpha -\beta }\mathrm {div}\varphi \), so that
by the semigroup property (2.4) of the Riesz potential. Moreover, in a similar way, we have
By Lemma 2.2, we thus have that \(\mathrm {div}^\alpha I_{\alpha -\beta }\varphi =\mathrm {div}^\beta \varphi \). Consequently, by Proposition 2.7, we get
Since \(|D^\alpha f|({\mathbb {R}}^n)<+\infty \), we have \(I_{\alpha -\beta }|D^\alpha f|\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) and thus, by Fubini’s Theorem, we get that
We conclude that
for any \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), proving (3.24).
We prove (3.25). By (3.24), by Tonelli’s Theorem and by [27, Lemma 2.4], we get
where \(C_{n,\alpha , U}\) is as in (2.9).
We now prove (3.26) in two steps. We argue as in the proof of (3.4).
Proof of (3.26), Step 1. Assume \(f\in C^\infty _c({\mathbb {R}}^n)\) and fix \(r>0\). We have
We estimate the two double integrals appearing in the right-hand side separately. By Tonelli’s Theorem, we have
Concerning the second double integral, we apply [1, Lemma 3.1.1(c)] to each component of the measure \(D^\alpha f\in {\mathscr {M}}({\mathbb {R}}^n;{\mathbb {R}}^n)\) and get
for all \(x\in A\). Since
we can compute
for all \(x\in A\). Hence, we have
Thus (3.4) follows for all \(f\in C^\infty _c({\mathbb {R}}^n)\) and \(r>0\).
Proof of (3.4), Step 2. Let \(f\in BV^\alpha ({\mathbb {R}}^n)\) and fix \(r>0\). By [27, Theorem 3.8], we find \((f_k)_{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^n)\) such that \(f_k\rightarrow f\) in \(L^1({\mathbb {R}}^n)\) and \(|D^\alpha f_k|({\mathbb {R}}^n)\rightarrow |D^\alpha f|({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \). By Step 1, we have that
for all \(k\in {\mathbb {N}}\). We have that
This can be proved arguing as in the proof of (3.9) using (3.25). At this point the proof goes like that of Proposition 3.2(i) and we thus leave the details to the reader. \(\square \)
4 Asymptotic behavior of fractional \(\alpha \)-variation as \(\alpha \rightarrow 1^-\)
4.1 Convergence of \(\nabla ^\alpha \) and \(\mathrm {div}^\alpha \) as \(\alpha \rightarrow 1^-\)
We begin with the following simple result about the asymptotic behavior of the constant \(\mu _{n,\alpha }\) as \(\alpha \rightarrow 1^-\).
Lemma 4.1
Let \(n\in {\mathbb {N}}\). We have
and
Proof
Since \(\Gamma (1) = 1\) and \(\Gamma (1 + x) = x \, \Gamma (x)\) for \(x > 0\) (see [9]), we have \(\Gamma (x) \sim x^{-1}\) as \(x \rightarrow 0^+\). Thus as \(\alpha \rightarrow 1^-\) we find
and (4.2) follows. Since \(\Gamma \) is log-convex on \((0,+\infty )\) (see [9]), for all \(x > 0\) and \(a \in (0, 1)\) we have
For \(x = \frac{n}{2}\) and \(a = \frac{\alpha + 1}{2}\), we can estimate
for all \(n\ge 1\). For \(x = 1 + \frac{1 - \alpha }{2}\) and \(a = \frac{\alpha }{2}\), we can estimate
We thus get
and (4.1) follows. \(\square \)
In the following technical result, we show that the constant \(C_{n, \alpha , U}\) defined in (2.9) is uniformly bounded as \(\alpha \rightarrow 1^{-}\) in terms of the volume and the diameter of the bounded open set \(U\subset {\mathbb {R}}^n\).
Lemma 4.2
(Uniform upper bound on \(C_{n, \alpha , U}\) as \(\alpha \rightarrow 1^-\)). Let \(n \in {\mathbb {N}}\) and \(\alpha \in (\frac{1}{2}, 1)\). Let \(U\subset {\mathbb {R}}^n\) be bounded open set. If \(C_{n, \alpha , U}\) is as in (2.9), then
where \(C_{n}\) is as in (4.1).
Proof
By (4.1), for all \(\alpha \in (\frac{1}{2},1)\) we have
Since \(t^{1 - \alpha } \le \max \{1, \sqrt{t}\}\) for any \(t\ge 0\) and \(\alpha \in (\frac{1}{2},1)\), we have
and
Combining these inequalities, we get the conclusion. \(\square \)
As consequence of Proposition 2.1 and Lemma 4.2, we prove that \(\nabla ^\alpha \) and \(\mathrm {div}^\alpha \) converge pointwise to \(\nabla \) and \(\mathrm {div}\) respectively as \(\alpha \rightarrow 1^{-}\).
Proposition 4.3
If \(f\in C^1_c({\mathbb {R}}^n)\), then for all \(x\in {\mathbb {R}}^n\) we have
As a consequence, if \(f \in C^{2}_{c}({\mathbb {R}}^{n})\) and \(\varphi \in C^{2}_c({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\), then for all \(x\in {\mathbb {R}}^n\) we have
Proof
Let \(f \in C^{1}_{c}({\mathbb {R}}^{n})\) and fix \(x\in {\mathbb {R}}^n\). Writing (2.6) in spherical coordinates, we find
Since \(f\in C^{1}_{c}({\mathbb {R}}^{n})\), for each fixed \(v\in \partial B_1\) we can integrate by parts in the variable \(\varrho \) and get
Clearly, we have
Thus, by Fubini’s Theorem, we conclude that
Since f has compact support and recalling (4.2), we can pass to the limit in (4.6) and get
proving (4.4). The pointwise limits in (4.5) immediately follows by Proposition 2.1. \(\square \)
In the following crucial result, we improve the pointwise convergence obtained in Proposition 4.3 to strong convergence in \(L^p({\mathbb {R}}^n)\) for all \(p\in [1,+\infty ]\).
Proposition 4.4
Let \(p\in [1,+\infty ]\). If \(f \in C^{2}_{c}({\mathbb {R}}^{n})\) and \(\varphi \in C^{2}_c({\mathbb {R}}^{n}; {\mathbb {R}}^{n})\), then
Proof
Let \(f \in C^{2}_{c}({\mathbb {R}}^{n})\). Since
for all \(x\in {\mathbb {R}}^n\) we can write
Therefore, by (2.6), we have
for all \(x\in {\mathbb {R}}^n\). We now distinguish two cases.
Case 1: \(p\in [1,+\infty )\). Using the elementary inequality \(|v+w|^p\le 2^{p-1}(|v|^p+|w|^p)\) valid for all \(v,w\in {\mathbb {R}}^n\), we have
We now estimate the two double integrals appearing in the right-hand side separately.
For the first double integral, as in the proof of Proposition 4.3, we pass in spherical coordinates to get
for all \(x\in {\mathbb {R}}^n\). Hence, by (4.2), we find
and
for all \(x\in {\mathbb {R}}^n\). Therefore, we get
for all \(x\in {\mathbb {R}}^n\). Recalling (4.1), we also observe that
for all \(\alpha \in (0,1)\), \(x\in {\mathbb {R}}^n\) and \(y\in B_1\). Moreover, letting \(R>0\) be such that \({{\,\mathrm{supp}\,}}f\subset B_R\), we can estimate
for all \(x\in {\mathbb {R}}^n\), so that
In conclusion, applying Lebesgue’s Dominated Convergence Theorem, we find
For the second double integral, note that
for all \(x\in {\mathbb {R}}^n\). Now let \(R>0\). Integrating by parts, we have that
for all \(x\in {\mathbb {R}}^n\). Since
and
for all \(R>0\), we conclude that
for all \(x\in {\mathbb {R}}^n\). Hence, by Minkowski’s Integral Inequality (see [76, Section A.1], for example), we can estimate
Thus, by (4.2), we get that
Case 2: \(p=+\infty \). We have
Again we estimate the two integrals appearing in the right-hand side separately. We note that
so that we can rewrite (4.7) as
Hence, we can estimate
so that
For the second integral, by (4.8) we can estimate
Thus, by (4.2), we get that
We can now conclude the proof. Again recalling (4.2), we thus find that
for all \(p\in [1,+\infty ]\) and the conclusion follows. The \(L^p\)-convergence of \(\mathrm {div}^\alpha \varphi \) to \(\mathrm {div}\varphi \) as \(\alpha \rightarrow 1^-\) for all \(p\in [1,+\infty ]\) follows by a similar argument and is left to the reader. \(\square \)
Remark 4.5
Note that the conclusion of Proposition 4.4 still holds if instead one assumes that \(f\in {\mathscr {S}}({\mathbb {R}}^n)\) and \(\varphi \in {\mathscr {S}}({\mathbb {R}}^n;{\mathbb {R}}^n)\), where \({\mathscr {S}}({\mathbb {R}}^n;{\mathbb {R}}^m)\) is the space of m-vector-valued Schwartz functions. We leave the proof of this assertion to the reader.
4.2 Weak convergence of \(\alpha \)-variation as \(\alpha \rightarrow 1^-\)
In Theorem 4.7 below, we prove that the fractional \(\alpha \)-variation weakly converges to the standard variation as \(\alpha \rightarrow 1^-\) for functions either in \(BV({\mathbb {R}}^{n})\) or in \(BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\cap L^{\infty }({\mathbb {R}}^{n})\). In the proof of Theorem 4.7, we are going to use the following technical result.
Lemma 4.6
There exists a dimensional constant \(c_n>0\) with the following property. If \(f \in L^{\infty }({\mathbb {R}}^{n})\cap BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\), then
for all \(R>0\) and \(\alpha \in (\frac{1}{2},1)\).
Proof
Since \(\Gamma (x)\sim x^{-1}\) as \(x\rightarrow 0^+\) (see [9]), inequality (4.9) follows immediately combining (3.7) with Lemma 4.1. \(\square \)
Theorem 4.7
If either \(f \in BV({\mathbb {R}}^{n})\) or \(f \in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\cap L^{\infty }({\mathbb {R}}^{n})\), then
Proof
We divide the proof in two steps.
Step 1. Assume \(f \in BV({\mathbb {R}}^{n})\). By [27, Theorem 3.18], we have
for all \(\varphi \in {{\,\mathrm{Lip}\,}}_c({\mathbb {R}}^n;{\mathbb {R}}^n)\). Thus, given \(\varphi \in C^2_c({\mathbb {R}}^n;{\mathbb {R}}^n)\), recalling Proposition 4.3 and the estimates (2.12) and (4.3), by Lebesgue’s Dominated Convergence Theorem we get that
To achieve the same limit for any \(\varphi \in C_c^0({\mathbb {R}}^n;{\mathbb {R}}^n)\), one just need to exploit (3.3) and the uniform estimate (4.3) in Lemma 4.2, and argue as in Step 2 of the proof of (3.4). We leave the details to the reader.
Step 2. Assume \(f \in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^{n})\cap L^{\infty }({\mathbb {R}}^{n})\). By Proposition 3.2(iii), we know that \(D^\alpha f=\nabla ^\alpha f{\mathscr {L}}^{n}\) with \(\nabla ^\alpha f\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n;{\mathbb {R}}^n)\). By Proposition 4.4, we get that
for all \(\varphi \in C^2_c({\mathbb {R}}^n;{\mathbb {R}}^n)\). To achieve the same limit for any \(\varphi \in C_c^0({\mathbb {R}}^n;{\mathbb {R}}^n)\), one just need to exploit (4.9) and argue as in Step 1. We leave the details to the reader. \(\square \)
We are now going to improve the weak convergence of the fractional \(\alpha \)-variation obtained in Theorem 4.7 by establishing the weak convergence also of the total fractional \(\alpha \)-variation as \(\alpha \rightarrow 1^-\), see Theorem 4.9 below. To do so, we need the following preliminary result.
Lemma 4.8
Let \(\mu \in {\mathscr {M}}({\mathbb {R}}^n; {\mathbb {R}}^m)\). We have \((I_{\alpha }\mu ){\mathscr {L}}^{n}\rightharpoonup \mu \) as \(\alpha \rightarrow 0^+\).
Proof
Since Riesz potential is a linear operator and thanks to Hahn–Banach Decomposition Theorem, without loss of generality we can assume that \(\mu \) is a nonnegative finite Radon measure.
Let now \(\varphi \in C_c^1({\mathbb {R}}^n)\) and let \(U\subset {\mathbb {R}}^n\) be a bounded open set such that \({{\,\mathrm{supp}\,}}\varphi \subset U\). We have that \(\Vert I_\alpha |\varphi |\Vert _{L^\infty ({\mathbb {R}}^n)}\le \kappa _{n,U}\Vert \varphi \Vert _{L^\infty ({\mathbb {R}}^n)}\) for all \(\alpha \in (0,\frac{1}{2})\) by [27, Lemma 2.4] and Lemma 4.2. Thus, by (4.4), Fubini’s Theorem and Lebesgue’s Dominated Convergence Theorem, we get that
To achieve the same limit for any \(\varphi \in C_c^0({\mathbb {R}}^n)\), one just need to exploit [27, Lemma 2.4] and (4.3) and argue as in Step 2 of the proof of (3.4). We leave the details to the reader. \(\square \)
Theorem 4.9
If either \(f\in BV({\mathbb {R}}^n)\) or \(f\in bv({\mathbb {R}}^n)\cap L^\infty ({\mathbb {R}}^n)\), then
Moreover, if \(f\in BV({\mathbb {R}}^n)\), then also
Proof
We prove (4.10) and (4.11) separately.
Proof of (4.10). By Theorem 4.7, we know that \(D^\alpha f\rightharpoonup Df\) as \(\alpha \rightarrow 1^-\). By [50, Proposition 4.29], we thus have that
for any open set \(A\subset {\mathbb {R}}^n\). Now let \(K\subset {\mathbb {R}}^n\) be a compact set. By the representation formula (3.18) in Corollary 3.6, we can estimate
Since \(|Df|({\mathbb {R}}^n)<+\infty \), by Lemma 4.8 and [50, Proposition 4.26] we can conclude that
and so (4.10) follows, thanks again to [50, Proposition 4.26].
Proof of (4.11). Now assume \(f\in BV({\mathbb {R}}^n)\). By (3.4) applied with \(A={\mathbb {R}}^n\) and \(r=1\), we have
By (4.2), we thus get that
Thus (4.11) follows by combining (4.12) for \(A={\mathbb {R}}^n\) with (4.13). \(\square \)
Remark 4.10
We notice that Theorems 4.7 and 4.9, in the case \(f=\chi _E\in BV({\mathbb {R}}^n)\) with \(E\subset {\mathbb {R}}^n\) bounded, and Theorem 4.11, were already announced in [71, Theorems 16 and 17].
Note that Theorems 4.7 and 4.9 in particular apply to any \(f\in W^{1,1}({\mathbb {R}}^n)\). In the following result, by exploiting Proposition 3.3, we prove that a stronger property holds for any \(f\in W^{1,p}({\mathbb {R}}^n)\) with \(p\in [1,+\infty )\).
Theorem 4.11
Let \(p\in [1,+\infty )\). If \(f\in W^{1,p}({\mathbb {R}}^n)\), then
Proof
By Proposition 3.3 we know that \(f\in S^{\alpha ,p}({\mathbb {R}}^n)\) for any \(\alpha \in (0, 1)\). We now assume \(p\in (1,+\infty )\) and divide the proof in two steps.
Step 1. We claim that
Indeed, on the one hand, by Proposition 4.4, we have
for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\), so that
for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\). We thus get that
On the other hand, applying (3.10) with \(A={\mathbb {R}}^n\) and \(r=1\), we have
By (4.2), we conclude that
Thus, (4.15) follows by combining (4.17) and (4.18).
Step 2. We now claim that
Indeed, let \(\varphi \in L^{\frac{p}{p-1}}({\mathbb {R}}^n;{\mathbb {R}}^n)\). For each \(\varepsilon >0\), let \(\psi _\varepsilon \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\) be such that \(\Vert \psi _\varepsilon -\varphi \Vert _{L^{\frac{p}{p-1}}({\mathbb {R}}^n;\,{\mathbb {R}}^n)}<\varepsilon \). By (4.16) and (4.15), we can estimate
so that (4.19) follows passing to the limit as \(\varepsilon \rightarrow 0^+\).
Since \(L^p({\mathbb {R}}^n;{\mathbb {R}}^n)\) is uniformly convex (see [19, Section 4.3] for example), the limit in (4.14) follows from (4.15) and (4.19) by [19, Proposition 3.32], and the proof in the case \(p\in (1,+\infty )\) is complete.
For the case \(p=1\), we argue as follows (we thank Mattia Calzi for this simple argument). Without loss of generality, it is enough to prove the limit in (4.15) with \(p=1\) for any given sequence \((\alpha _k)_{k\in {\mathbb {N}}}\) such that \(\alpha _k\rightarrow 1^-\) as \(k\rightarrow +\infty \). By (4.11), the sequence \((\Vert \nabla ^{\alpha _k} f\Vert _{L^1({\mathbb {R}}^n;\,{\mathbb {R}}^n)})_{k\in {\mathbb {N}}}\) is bounded for any \(f\in W^{1,1}({\mathbb {R}}^n)\) and thus, by Banach–Steinhaus Theorem, the linear operators \(\nabla ^{\alpha _k}:W^{1,1}({\mathbb {R}}^n)\rightarrow L^1({\mathbb {R}}^n;{\mathbb {R}}^n)\), \(k\in {\mathbb {N}}\), are uniformly bounded (in the operator norm). The conclusion hence follows by exploiting the density of \(C^\infty _c({\mathbb {R}}^n)\) in \(W^{1,1}({\mathbb {R}}^n)\) and Proposition 4.4. \(\square \)
For the case \(p=+\infty \), we have the following result. The proof is very similar to the one of Theorem 4.11 and is thus left to the reader.
Theorem 4.12
If \(f\in W^{1,\infty }({\mathbb {R}}^n)\), then
and
4.3 \(\Gamma \)-convergence of \(\alpha \)-variation as \(\alpha \rightarrow 1^-\)
In this section, we study the \(\Gamma \)-convergence of the fractional \(\alpha \)-variation to the standard variation as \(\alpha \rightarrow 1^-\).
We begin with the \(\Gamma \text { -}\liminf \) inequality.
Theorem 4.13
(\(\Gamma \text { -}\liminf \) inequalities as \(\alpha \rightarrow 1^-\)) Let \(\Omega \subset {\mathbb {R}}^n\) be an open set.
-
(i)
If \((f_\alpha )_{\alpha \in (0,1)}\subset L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) satisfies \(\sup _{\alpha \in (0,1)}\Vert f_\alpha \Vert _{L^\infty ({\mathbb {R}}^n)}<+\infty \) and \(f_\alpha \rightarrow f\) in \(L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) as \(\alpha \rightarrow 1^-\), then
$$\begin{aligned} |Df|(\Omega )\le \liminf \limits _{\alpha \rightarrow 1^-}|D^\alpha f_\alpha |(\Omega ). \end{aligned}$$(4.22) -
(ii)
If \((f_\alpha )_{\alpha \in (0,1)}\subset L^1({\mathbb {R}}^n)\) satisfies \(f_\alpha \rightarrow f\) in \(L^1({\mathbb {R}}^n)\) as \(\alpha \rightarrow 1^-\), then (4.22) holds.
Proof
We prove the two statements separately.
Proof of (i). Let \(\varphi \in C^\infty _c(\Omega ;{\mathbb {R}}^n)\) be such that \(\Vert \varphi \Vert _{L^\infty (\Omega ;{\mathbb {R}}^n)}\le 1\). Since we can estimate
by Proposition 4.4 we get that
and the conclusion follows.
Proof of (ii). Let \(\varphi \in C^\infty _c(\Omega ;{\mathbb {R}}^n)\) be such that \(\Vert \varphi \Vert _{L^\infty (\Omega ;{\mathbb {R}}^n)}\le 1\). Since we can estimate
by Proposition 4.4 we get that
and the conclusion follows. \(\square \)
We now pass to the \(\Gamma \text { -}\limsup \) inequality.
Theorem 4.14
(\(\Gamma \text { -}\limsup \) inequalities as \(\alpha \rightarrow 1^-\)) Let \(\Omega \subset {\mathbb {R}}^n\) be an open set.
-
(i)
If \(f\in BV({\mathbb {R}}^n)\) and either \(\Omega \) is bounded or \(\Omega ={\mathbb {R}}^n\), then
$$\begin{aligned} \limsup _{\alpha \rightarrow 1^-}|D^\alpha f|(\Omega )\le |Df|({\overline{\Omega }}). \end{aligned}$$(4.23) -
(ii)
If \(f\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) and \(\Omega \) is bounded, then
$$\begin{aligned} \Gamma (L^1_{{{\,\mathrm{loc}\,}}})\text { -}\limsup _{\alpha \rightarrow 1^-}|D^\alpha f|(\Omega )\le |Df|({\overline{\Omega }}). \end{aligned}$$
In addition, if \(f=\chi _E\), then the recovering sequences \((f_\alpha )_{\alpha \in (0,1)}\) in (i) and (ii) can be taken such that \(f_\alpha =\chi _{E_\alpha }\) for some measurable sets \((E_\alpha )_{\alpha \in (0,1)}\).
Proof
Assume \(f\in BV({\mathbb {R}}^n)\). By Theorem 4.9, we know that \(|D^\alpha f|\rightharpoonup |Df|\) as \(\alpha \rightarrow 1^-\). Thus, by [50, Proposition 4.26], we get that
for any bounded open set \(\Omega \subset {\mathbb {R}}^n\). If \(\Omega ={\mathbb {R}}^n\), then (4.23) follows immediately from (4.11). This concludes the proof of (i).
Now assume that \(f\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) and \(\Omega \) is bounded. Let \((R_k)_{k\in {\mathbb {N}}}\subset (0,+\infty )\) be a sequence such that \(R_k\rightarrow +\infty \) as \(k\rightarrow +\infty \) and set \(f_k:=f\chi _{B_{R_k}}\) for all \(k\in {\mathbb {N}}\). By Theorem A.1, we can choose the sequence \((R_k)_{k\in {\mathbb {N}}}\) such that, in addition, \(f_k\in BV({\mathbb {R}}^n)\) with \(Df_k=\chi _{B_{R_k}}^\star Df+f^\star D\chi _{B_{R_k}}\) for all \(k\in {\mathbb {N}}\). Consequently, \(f_k\rightarrow f\) in \(L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \) and, moreover, since \(\Omega \) is bounded, \(|Df_k|(\Omega )=|Df|(\Omega )\) and \(|Df_k|(\partial \Omega )=|Df|(\partial \Omega )\) for all \(k\in {\mathbb {N}}\) sufficiently large. By (4.24), we have that
for all \(k\in {\mathbb {N}}\) sufficiently large. Hence, by [17, Proposition 1.28], by [29, Proposition 8.1(c)] and by (4.25), we get that
This concludes the proof of (ii).
Finally, if \(f=\chi _E\), then we can repeat the above argument verbatim in the metric spaces \(\{\chi _F\in L^1({\mathbb {R}}^n): F\subset {\mathbb {R}}^n\}\) for (i) and \(\{\chi _F\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n): F\subset {\mathbb {R}}^n\}\) for (ii) endowed with their natural distances. \(\square \)
Remark 4.15
Thanks to (4.23), a recovery sequence in Theorem 4.14(i) is the constant sequence (also in the special case \(f = \chi _{E}\)).
Combining Theorems 4.13(i) and 4.14(ii), we can prove that the fractional Caccioppoli \(\alpha \)-perimeter \(\Gamma \)-converges to De Giorgi’s perimeter as \(\alpha \rightarrow 1^-\) in \(L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\). We refer to [3] for the same result on the classical fractional perimeter.
Theorem 4.16
(\(\Gamma (L^1_{{{\,\mathrm{loc}\,}}})\text { -}\lim \) of perimeters as \(\alpha \rightarrow 1^-\)) Let \(\Omega \subset {\mathbb {R}}^n\) be a bounded open set with Lipschitz boundary. For every measurable set \(E\subset {\mathbb {R}}^n\), we have
Proof
By Theorem 4.13(i), we already know that
so we just need to prove the \(\Gamma (L^1_{{{\,\mathrm{loc}\,}}})\text { -}\limsup \) inequality. Without loss of generality, we can assume \(P(E;\Omega )<+\infty \). Now let \((E_k)_{k\in {\mathbb {N}}}\) be given by Theorem A.4. Since \(\chi _{E_k}\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) and \(P(E_k;\partial \Omega )=0\) for all \(k\in {\mathbb {N}}\), by Theorem 4.14(ii) we know that
for all \(k\in {\mathbb {N}}\). Since \(\chi _{E_k}\rightarrow \chi _E\) in \(L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) and \(P(E_k;\Omega )\rightarrow P(E;\Omega )\) as \(k\rightarrow +\infty \), by [17, Proposition 1.28] we get that
and the proof is complete. \(\square \)
Finally, by combining Theorems 4.13(ii) and 4.14, we can prove that the fractional \(\alpha \)-variation \(\Gamma \)-converges to De Giorgi’s variation as \(\alpha \rightarrow 1^-\) in \(L^1({\mathbb {R}}^n)\).
Theorem 4.17
(\(\Gamma (L^1)\text { -}\lim \) of variations as \(\alpha \rightarrow 1^-\)) Let \(\Omega \subset {\mathbb {R}}^n\) be an open set such that either \(\Omega \) is bounded with Lipschitz boundary or \(\Omega ={\mathbb {R}}^n\). For every \(f\in BV({\mathbb {R}}^n)\), we have
Proof
The case \(\Omega ={\mathbb {R}}^n\) follows immediately by [29, Proposition 8.1(c)] combining Theorem 4.13(ii) with Theorem 4.14(i). We can thus assume that \(\Omega \) is a bounded open set with Lipschitz boundary and argue as in the proof of Theorem 4.16. By Theorem 4.13(ii), we already know that
so we just need to prove the \(\Gamma (L^1)\text { -}\limsup \) inequality. Without loss of generality, we can assume \(|Df|(\Omega )<+\infty \). Now let \((f_k)_{k\in {\mathbb {N}}}\subset BV({\mathbb {R}}^n)\) be given by Theorem A.6. Since \(|Df_k|(\partial \Omega )=0\) for all \(k\in {\mathbb {N}}\), by Theorem 4.14 we know that
for all \(k\in {\mathbb {N}}\). Since \(f_k\rightarrow f\) in \(L^1({\mathbb {R}}^n)\) and \(|D^\alpha f_k|(\Omega )\rightarrow |D^\alpha f|(\Omega )\) as \(k\rightarrow +\infty \), by [17, Proposition 1.28] we get that
and the proof is complete. \(\square \)
Remark 4.18
Thanks to Theorem 4.17, we can slightly improve Theorem 4.16. Indeed, if \(\chi _{E} \in BV({\mathbb {R}}^{n})\), then we also have
for any open set \(\Omega \subset {\mathbb {R}}^n\) such that either \(\Omega \) is bounded with Lipschitz boundary or \(\Omega ={\mathbb {R}}^n\).
5 Asymptotic behavior of fractional \(\beta \)-variation as \(\beta \rightarrow \alpha ^-\)
5.1 Convergence of \(\nabla ^\beta \) and \(\mathrm {div}^\beta \) as \(\beta \rightarrow \alpha \)
We begin with the following simple result about the \(L^1\)-convergence of the operators \(\nabla ^\beta \) and \(\mathrm {div}^\beta \) as \(\beta \rightarrow \alpha \) with \(\alpha \in (0,1)\).
Lemma 5.1
Let \(\alpha \in (0,1)\). If \(f\in W^{\alpha ,1}({\mathbb {R}}^n)\) and \(\varphi \in W^{\alpha ,1}({\mathbb {R}}^n;{\mathbb {R}}^n)\), then
Proof
Given \(\beta \in (0,\alpha )\), we can estimate
Since the \(\Gamma \) function is continuous (see [9]), we clearly have
Now write
On the one hand, since \(f\in W^{\alpha ,1}({\mathbb {R}}^n)\), we have
and thus, by Lebesgue’s Dominated Convergence Theorem, we get that
On the other hand, since one has
for all \(\beta \in (0,\alpha )\), we can estimate
for all \(\beta \in \left( \frac{\alpha }{2},\alpha \right) \) and thus, by Lebesgue’s Dominated Convergence Theorem, we get that
and the first limit in (5.1) follows. The second limit in (5.1) follows similarly and we leave the proof to the reader. \(\square \)
Remark 5.2
Let \(\alpha \in (0,1)\). If \(f\in W^{\alpha +\varepsilon ,1}({\mathbb {R}}^n)\) and \(\varphi \in W^{\alpha +\varepsilon ,1}({\mathbb {R}}^n)\) for some \(\varepsilon \in (0,1-\alpha )\), then, arguing as in the proof of Lemma 5.1, one can also prove that
We leave the details of proof of this result to the interested reader.
If one deals with more regular functions, then Lemma 5.1 can be improved as follows.
Lemma 5.3
Let \(\alpha \in (0,1)\) and \(p \in [1, + \infty ]\). If \(f\in {{\,\mathrm{Lip}\,}}_{c}({\mathbb {R}}^n)\) and \(\varphi \in {{\,\mathrm{Lip}\,}}_{c}({\mathbb {R}}^n;{\mathbb {R}}^n)\), then
Proof
Since clearly \(f \in W^{\alpha , 1}({\mathbb {R}}^{n})\) for any \(\alpha \in (0, 1)\), the first limit in (5.2) for the case \(p=1\) follows from Lemma 5.1. Hence, we just need to prove the validity of the same limit for the case \(p = + \infty \), since then the conclusion simply follows by an interpolation argument.
Let \(\beta \in (0, \alpha )\) and \(x \in {\mathbb {R}}^{n}\). We have
Since
and
for all \(\beta \in \big (\frac{\alpha }{2}, \alpha \big )\) we obtain
for some constant \(c_{n, \alpha } > 0\) depending only on n and \(\alpha \). Thus the conclusion follows since \(\mu _{n,\beta }\rightarrow \mu _{n,\alpha }\) as \(\beta \rightarrow \alpha ^-\). The second limit in (5.2) follows similarly and we leave the proof to the reader. \(\square \)
5.2 Weak convergence of \(\beta \)-variation as \(\beta \rightarrow \alpha ^-\)
In Theorem 5.4 below, we prove the weak convergence of the \(\beta \)-variation as \(\beta \rightarrow \alpha ^-\). The proof is very similar to those of Theorem 4.7 and Theorem 4.9 and is thus left to the reader.
Theorem 5.4
Let \(\alpha \in (0,1)\). If \(f \in BV^\alpha ({\mathbb {R}}^{n})\), then
Moreover, we have
5.3 \(\Gamma \)-convergence of \(\beta \)-variation as \(\beta \rightarrow \alpha ^-\)
In this section, we study the \(\Gamma \)-convergence of the fractional \(\beta \)-variation as \(\beta \rightarrow \alpha ^-\), partially extending the results obtained in Sect. 4.3.
We begin with the \(\Gamma \text { -}\liminf \) inequality.
Theorem 5.5
(\(\Gamma \text { -}\liminf \) inequality for \(\beta \rightarrow \alpha ^-\)) Let \(\alpha \in (0,1)\) and let \(\Omega \subset {\mathbb {R}}^n\) be an open set. If \((f_\beta )_{\beta \in (0,\alpha )}\subset L^1({\mathbb {R}}^n)\) satisfies \(f_\beta \rightarrow f\) in \(L^1({\mathbb {R}}^n)\) as \(\beta \rightarrow \alpha ^-\), then
Proof
We argue as in the proof of Theorem 4.13(ii). Let \(\varphi \in C^\infty _c(\Omega ;{\mathbb {R}}^n)\) be such that \(\Vert \varphi \Vert _{L^\infty (\Omega ;{\mathbb {R}}^n)}\le 1\). Let \(U\subset {\mathbb {R}}^n\) be a bounded open set such that \({{\,\mathrm{supp}\,}}\varphi \subset U\). By (2.12), we can estimate
for all \(\beta \in (0,\alpha )\). Since \(\mathrm {div}^\beta \varphi \rightarrow \mathrm {div}^\alpha \varphi \) in \(L^{\infty }({\mathbb {R}}^{n})\) as \(\beta \rightarrow \alpha ^-\) by (5.2), we easily obtain
Hence, we get
and the conclusion follows. \(\square \)
We now pass to the \(\Gamma \text { -}\limsup \) inequality.
Theorem 5.6
(\(\Gamma \text { -}\limsup \) inequality for \(\beta \rightarrow \alpha ^-\)) Let \(\alpha \in (0,1)\) and let \(\Omega \subset {\mathbb {R}}^n\) be an open set. If \(f\in BV^\alpha ({\mathbb {R}}^n)\) and either \(\Omega \) is bounded or \(\Omega ={\mathbb {R}}^n\), then
Proof
We argue as in the proof of Theorem 4.14. By Theorem 5.4, we know that \(|D^\beta f|\rightharpoonup |D^\alpha f|\) as \(\beta \rightarrow \alpha ^-\). Thus, by [50, Proposition 4.26] and (5.3), we get that
for any open set \(\Omega \subset {\mathbb {R}}^n\) such that either \(\Omega \) is bounded or \(\Omega ={\mathbb {R}}^n\). \(\square \)
Corollary 5.7
(\(\Gamma (L^1)\text { -}\lim \) of variations in \({\mathbb {R}}^n\) as \(\beta \rightarrow \alpha ^-\)) Let \(\alpha \in (0, 1)\). For every \(f\in BV^{\alpha }({\mathbb {R}}^n)\), we have
In particular, the constant sequence is a recovery sequence.
Proof
The result follows easily by combining (5.4) and (5.5) in the case \(\Omega = {\mathbb {R}}^{n}\). \(\square \)
Remark 5.8
We recall that, by [27, Theorem 3.25], \(f \in BV^{\alpha }({\mathbb {R}}^{n})\) satisfies \(|D^{\alpha } f| \ll {\mathscr {L}}^{n}\) if and only if \(f \in S^{\alpha , 1}({\mathbb {R}}^{n})\). Therefore, if \(f \in S^{\alpha , 1}({\mathbb {R}}^{n})\), then \(|D^{\alpha }f|(\partial \Omega ) = 0\) for any bounded open set \(\Omega \subset {\mathbb {R}}^n\) such that \({\mathscr {L}}^{n}(\partial \Omega ) = 0\) (for instance, \(\Omega \) with Lipschitz boundary). Thus, we can actually obtain the \(\Gamma \)-convergence of the fractional \(\beta \)-variation as \(\beta \rightarrow \alpha ^-\) on bounded open sets with Lipschitz boundary for any \(f \in S^{\alpha , 1}({\mathbb {R}}^{n})\) too. Indeed, it is enough to combine (5.4) and (5.5) and then exploit the fact that \(|D^{\alpha } f|(\partial \Omega )=0\) to get
for any \(f \in S^{\alpha , 1}({\mathbb {R}}^{n})\).
We were not able to find a reference for the analogue of Corollary 5.7 for the usual fractional Sobolev seminorms. For the sake of completeness, we state and prove it below for all \(p\in [1,+\infty )\) on a general open set.
Theorem 5.9
(\(\Gamma (L^p)\text { -}\lim \) of \(W^{\beta , p}\)-seminorm as \(\beta \rightarrow \alpha ^-\)) Let \(\Omega \subset {\mathbb {R}}^n\) be a non-empty open set, \(\alpha \in (0, 1)\) and \(p\in [1,+\infty )\). For every \(f\in W^{\alpha , p}(\Omega )\), we have
In particular, the constant sequence is a recovery sequence.
Proof
Let \((f_\beta )_{\beta \in (0,\alpha )}\subset L^p(\Omega )\) be such that \(f_\beta \rightarrow f\) in \(L^p(\Omega )\) as \(\beta \rightarrow \alpha ^-\). Let \((\beta _k) \subset (0, \alpha )\) be such that \(\beta _k\rightarrow \alpha \) as \(k\rightarrow +\infty \) and
Up to extract a further subsequence, we can assume that \(f_{\beta _k}(x) \rightarrow f(x)\) as \(k \rightarrow + \infty \) for a.e. \(x \in \Omega \). Then we can estimate
by Fatou’s Lemma. We thus get that
Since
by the Monotone Convergence Theorem, we also have that
and the conclusion immediately follows. \(\square \)
Change history
27 August 2022
Missing Open Access funding information has been added in the Funding Note.
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Acknowledgements
The authors thank Luigi Ambrosio, Elia Brué, Mattia Calzi, Quoc-Hung Nguyen and Daniel Spector for many valuable suggestions and useful comments. The authors also wish to express their gratitude to the anonymous referees for their insightful remarks. This research was partially supported by the PRIN2015 MIUR Project “Calcolo delle Variazioni”. The second author is a member of INdAM–GNAMPA and is partially supported by the ERC Starting Grant 676675 FLIRT—Fluid Flows and Irregular Transport and by the INdAM–GNAMPA Project 2020 Problemi isoperimetrici con anisotropie (n. prot. U-UFMBAZ-2020-000798 15-04-2020).
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Appendix A. Truncation and approximation of BV functions
Appendix A. Truncation and approximation of BV functions
In this appendix, we deal with two results on BV functions and sets with locally finite perimeter. These results are well known to experts, but we decided to state and prove them here because either we were not able to find them formulated in the exact form we needed or the results available in the literature were not proved in full correctness (see Remark A.5 below).
1.1 A.1 Truncation of BV functions
Following [4, Section 3.6] and [34, Section 5.9], given \(f\in L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\), we define its precise representative \(f^\star :{\mathbb {R}}^n\rightarrow [-\infty ,+\infty ]\) as
if the limit exists, otherwise we let \(f^\star (x)=0\) by convention.
Theorem A.1
(Truncation of BV functions) If \(f\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\), then
for \({\mathscr {L}}^{1}\)-a.e. \(r>0\). If, in addition, \(f\in L^\infty ({\mathbb {R}}^n)\), then (A.2) holds for all \(r>0\).
Proof
Fix \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\) and let \(U\subset {\mathbb {R}}^n\) be a bounded open set such that \({{\,\mathrm{supp}\,}}(\varphi )\subset U\). Let \((\varrho _\varepsilon )_{\varepsilon >0}\subset C^\infty _c({\mathbb {R}}^n)\) be a family of standard mollifiers as in [27, Section 3.3] and set \(f_\varepsilon :=f*\varrho _\varepsilon \) for all \(\varepsilon >0\). Note that \({{\,\mathrm{supp}\,}}\big (\varrho _\varepsilon *(\chi _{B_r}\varphi )\big )\subset U\) and \({{\,\mathrm{supp}\,}}\big (\varrho _\varepsilon *(\chi _{B_r}\mathrm {div}\varphi )\big )\subset U\) for all \(\varepsilon >0\) sufficiently small and for all \(r>0\). Given \(r>0\), by Leibniz’s rule and Fubini’s Theorem, we have
Since \(f_\varepsilon \rightarrow f\) a.e. in \({\mathbb {R}}^n\) as \(\varepsilon \rightarrow 0^+\) and
for all \(\varepsilon >0\), by Lebesgue’s Dominated Convergence Theorem we have
for all \(r>0\). Thus, since \(\varrho _\varepsilon *(\chi _{B_r}\varphi )\rightarrow \chi _{B_r}^\star \varphi \) pointwise in \({\mathbb {R}}^n\) as \(\varepsilon \rightarrow 0^+\) and
for all \(\varepsilon >0\) sufficiently small, again by Lebesgue’s Dominated Convergence Theorem we have
for all \(r>0\). Now, by [4, Theorem 3.78 and Corollary 3.80], we know that \(f_\varepsilon \rightarrow f^\star \) \({\mathscr {H}}^{n-1}\)-a.e. in \({\mathbb {R}}^n\) as \(\varepsilon \rightarrow 0^+\). As a consequence, given any \(r>0\), we get that \(f_\varepsilon \rightarrow f^\star \) \(|D\chi _{B_r}|\)-a.e. in \({\mathbb {R}}^n\) as \(\varepsilon \rightarrow 0^+\). Thus, if \(f\in L^\infty ({\mathbb {R}}^n)\), then
for all \(\varepsilon >0\) and so, again by Lebesgue’s Dominated Convergence Theorem, we have
for all \(r>0\). Therefore, if \(f\in L^\infty ({\mathbb {R}}^n)\), then we can pass to the limit as \(\varepsilon \rightarrow 0^+\) in (A.3) and get
for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\) and for all \(r>0\). Since \(\Vert f^\star \Vert _{L^\infty ({\mathbb {R}}^n)}\le \Vert f\Vert _{L^\infty ({\mathbb {R}}^n)}\), this proves (A.2) for all \(r>0\).
If f is not necessarily bounded, then we argue as follows. We start by observing that, since
the set
satisfies \({\mathscr {L}}^{1}((0,+\infty )\setminus W)=0\). Without loss of generality, assume that \(\Vert \varphi \Vert _{L^\infty ({\mathbb {R}}^n;\,{\mathbb {R}}^n)}\le 1\). Hence, for all \(r \in W\), we can estimate
Given any \(R > 0\), by Fatou’s Lemma we thus get that
Hence, the set
satisfies \({\mathscr {L}}^{1}((0,+\infty )\setminus Z)=0\) and clearly does not depend on the choice of \(\varphi \). Now fix \(r\in Z \cap W\) and let \((\varepsilon _k)_{k\in {\mathbb {N}}}\) be any sequence realising the \(\liminf \) in (A.5). By (A.4), we thus get
uniformly for all \(\varphi \) satisfying \(\Vert \varphi \Vert _{L^\infty ({\mathbb {R}}^n;\,{\mathbb {R}}^n)}\le 1\). Passing to the limit along the sequence \((\varepsilon _k)_{k\in {\mathbb {N}}}\) as \(k\rightarrow +\infty \) in (A.3), we get that
for all \(\varphi \in C^\infty _c({\mathbb {R}}^n;{\mathbb {R}}^n)\) with \(\Vert \varphi \Vert _{L^\infty ({\mathbb {R}}^n;\,{\mathbb {R}}^n)}\le 1\). Thus (A.2) follows for all \(r\in W\cap Z\) and the proof is concluded. \(\square \)
1.2 A.2 Approximation by sets with polyhedral boundary
In this section we state and prove standard approximation results for sets with finite perimeter or, more generally, \(BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) functions, in a sufficiently regular bounded open set.
We need the following two preliminary lemmas.
Lemma A.2
Let \(V,W\subset {\mathbb {S}}^{n-1}\), with V finite and W at most countable. For any \(\varepsilon >0\), there exists \({\mathcal {R}}\in \mathrm {SO}(n)\) with \(|{\mathcal {R}}-{\mathcal {I}}|<\varepsilon \), where \({\mathcal {I}}\) is the identity matrix, such that \({\mathcal {R}}(V)\cap W=\varnothing \).
Proof
Let \(N\in {\mathbb {N}}\) be such that \(V=\Bigg \{v_i\in {\mathbb {S}}^{n-1} : i=1,\dots ,N\Bigg \}\). We divide the proof in two steps.
Step 1. Assume that W is finite and set \(A_i:=\Bigg \{{\mathcal {R}}\in \mathrm {SO}(n) : {\mathcal {R}}(v_i)\notin W\Bigg \}\) for all \(i=1,\dots ,N\). We now claim that \(A_i\) of \(\mathrm {SO}(n)\) for all \(i=1,\dots ,N\). Indeed, given any \(i=1,\dots ,N\), since W is finite, the set \(A_i^c=\mathrm {SO}(n)\setminus A_i\) is closed in \(\mathrm {SO}(n)\). Moreover, we claim that \(\mathrm {int}(A_i^c) = \varnothing \). Indeed, by contradiction, let us assume that \(\mathrm {int}(A_i^c)\ne \varnothing \). Then there exist \(\varepsilon >0\) and \({\mathcal {R}}\in A_i^c\) such that any \({\mathcal {S}}\in \mathrm {SO}(n)\) with \(|{\mathcal {S}}-{\mathcal {R}}|<\varepsilon \) satisfies \({\mathcal {S}}\in A_i^c\). Now let
and define \({\mathcal {S}}_\vartheta := {\mathcal {Q}}_\vartheta \,\mathcal R\in \mathrm {SO}(n)\) for all \(\vartheta \in [0,2\pi )\). Since
for all \(\vartheta \in [0,\delta ]\) for some \(\delta >0\) depending only on \(\varepsilon \) and \({\mathcal {R}}\), we get that \({\mathcal {S}}_\vartheta \in A_i^c\) for all \(\vartheta \in [0,\delta ]\). Therefore \({\mathcal {S}}_\vartheta (v_i)\in W\) for all \(\vartheta \in [0,\delta ]\), in contrast with the fact that W is finite. Thus, \(A_i\) is an open and dense subset of \(\mathrm {SO}(n)\) for all \(i=1,\dots ,N\), and so also the set
is an open and dense subset of \(\mathrm {SO}(n)\). The result is thus proved for any finite set W.
Step 2. Now assume that W is countable, \(W=\{w_k\in {\mathbb {S}}^{n-1} : k\in {\mathbb {N}}\}\). For all \(M\in {\mathbb {N}}\), set \(W_M:=\{w_k\in W : k\le M\}\). By Step 1, we know that \(A^{W_M}\) is an open and dense subset of \(\mathrm {SO}(n)\) for all \(M\in {\mathbb {N}}\). Since \(\mathrm {SO}(n)\subset {\mathbb {R}}^{n^2}\) is compact, by Baire’s Theorem \(A:=\bigcap _{M\in {\mathbb {N}}} A^{W_M}\) is a dense subset of \(\mathrm {SO}(n)\). This concludes the proof. \(\square \)
Since \(\det :\mathrm {GL}(n)\rightarrow {\mathbb {R}}\) is a continuous map, there exists a dimensional constant \(\delta _n\in (0,1)\) such that \(\det {\mathcal {R}}\ge \frac{1}{2}\) for all \({\mathcal {R}}\in \mathrm {GL}(n)\) with \(|{\mathcal {R}}-{\mathcal {I}}|<\delta _n\).
Lemma A.3
Let \(\varepsilon \in (0,\delta _n)\) and let \(E\subset {\mathbb {R}}^n\) be a bounded set with \(P(E)<+\infty \). If \({\mathcal {R}}\in \mathrm {SO}(n)\) satisfies \(|{\mathcal {R}}-{\mathcal {I}}|<\varepsilon \), then
where \(r_E:=\sup \{r>0 : |E\setminus B_r|>0\}\).
Proof
We divide the proof in two steps.
Step 1. Let \(r>0\) and let \(f\in C^\infty _c({\mathbb {R}}^n)\). Setting \({\mathcal {R}}_t:=(1-t){\mathcal {I}}+t{\mathcal {R}}\) for all \(t\in [0,1]\), we can estimate
Since \(|{\mathcal {R}}_t-{\mathcal {I}}|=t|{\mathcal {R}}-{\mathcal {I}}|<t\varepsilon <\delta _n\) for all \(t\in [0,1]\), \({\mathcal {R}}_t\) is invertible with \(\det ({\mathcal {R}}_t^{-1})\le 2\) for all \(t\in [0,1]\). Hence we can estimate
so that
Step 2. Since \(\chi _E\in BV({\mathbb {R}}^n)\), by combining [34, Theorem 5.3] with a standard cut-off approximation argument, we find \((f_k)_{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^n)\) such that \(f_k\rightarrow \chi _E\) pointwise a.e. in \({\mathbb {R}}^n\) and \(|\nabla f_k|({\mathbb {R}}^n)\rightarrow P(E)\) as \(k\rightarrow +\infty \). Given any \(r>0\), by (A.6) in Step 1 we have
for all \(k\in {\mathbb {N}}\). Passing to the limit as \(k\rightarrow +\infty \), by Fatou’s Lemma we get that
Since \(E\subset B_{r_E}\) up to \({\mathscr {L}}^{n}\)-negligible sets, also \({\mathcal {R}}(E)\subset B_{r_E}\) up to \({\mathscr {L}}^{n}\)-negligible sets. Thus we can choose \(r=r_E\) and the proof is complete. \(\square \)
We are now ready to prove the main approximation result, see also [3, Proposition 15].
Theorem A.4
Let \(\Omega \subset {\mathbb {R}}^n\) be a bounded open set with Lipschitz boundary and let \(E\subset {\mathbb {R}}^n\) be a measurable set such that \(P(E;\Omega )<+\infty \). There exists a sequence \((E_k)_{k\in {\mathbb {N}}}\) of bounded open sets with polyhedral boundary such that
for all \(k\in {\mathbb {N}}\) and
as \(k\rightarrow +\infty \).
Proof
We divide the proof in four steps.
Step 1: cut-off. Since \(\Omega \) is bounded, we find \(R_0>0\) such that \({\overline{\Omega }}\subset B_{R_0}\). Let us define \(R_k=R_0+k\) and
for all \(k\in {\mathbb {N}}\). We set \(E^1_k:=E\cap B_{R_k}\cap C_k^c\) for all \(k\in {\mathbb {N}}\). Note that \(E_k^1\) is a bounded measurable set such that
and
Step 2: extension. Let us define
for all \(k\in {\mathbb {N}}\). Since \(\chi _{E_k^1\cap \Omega }\in BV(\Omega )\) for all \(k\in {\mathbb {N}}\), by [4, Definition 3.20 and Proposition 3.21] there exists a sequence \((v_k)_{k\in {\mathbb {N}}}\subset BV({\mathbb {R}}^n)\) such that
for all \(k\in {\mathbb {N}}\). Let us define \(F^t_k:=\{v_k>t\}\) for all \(t\in (0,1)\). Given \(k\in {\mathbb {N}}\), by the coarea formula [4, Theorem 3.40], for a.e. \(t\in (0,1)\) the set \(F_k^t\) has finite perimeter in \({\mathbb {R}}^n\) and satisfies
for all \(k\in {\mathbb {N}}\). We choose any such \(t_k\in (0,1)\) for each \(k\in {\mathbb {N}}\) and define \(E_k^2:=E_k^1\cup F_k^{t_k}\) for all \(k\in {\mathbb {N}}\). Note that \(E_k^2\) is a bounded set with finite perimeter in \({\mathbb {R}}^n\) such that
and
Step 3: approximation. Let us define
for all \(k\in {\mathbb {N}}\). First arguing as in the first part of the proof of [50, Theorem 13.8] taking [50, Remark 13.13] into account, and then performing a standard diagonal argument, we find a sequence of bounded open sets \((E_k^3)_{k\in {\mathbb {N}}}\) with polyhedral boundary such that
and
as \(k\rightarrow +\infty \). If there exists a subsequence \((E_{k_j}^3)_{j\in {\mathbb {N}}}\) such that \(P(E_{k_j}^3;\partial \Omega )=0\) for all \(j\in {\mathbb {N}}\), then we can set \(E_j:=E_{k_j}\) for all \(j\in {\mathbb {N}}\) and the proof is concluded. If this is not the case, then we need to proceed with the next last step.
Step 4: rotation. We now argue as in the last part of the proof of [3, Proposition 15]. Fix \(k\in {\mathbb {N}}\) and assume \(P(E^3_k;\partial \Omega )>0\). Since \(E_k^3\) has polyhedral boundary, we have \({\mathscr {H}}^{n-1}(\partial E^3_k\cap \partial \Omega )>0\) if and only if there exist \(\nu \in {\mathbb {S}}^{n-1}\) and \(U\subset {\mathscr {F}}\Omega \) such that \({\mathscr {H}}^{n-1}(U)>0\), \(\nu _\Omega (x)=\nu \) for all \(x\in U\) and \(U\subset \partial H\) for some (affine) half-space H satisfying \(\nu _H=\nu \). Since \(P(\Omega )={\mathscr {H}}^{n-1}(\partial \Omega )<+\infty \), the set
is at most countable. Since \(E_k^3\) has polyhedral boundary, the set
is finite. By Lemma A.2, given \(\varepsilon _k>0\), there exists \({\mathcal {R}}_k\in \mathrm {SO}(n)\) with \(|{\mathcal {R}}_k-{\mathcal {I}}|<\varepsilon _k\) such that \({\mathcal {R}}_k(V_k)\cap W=\varnothing \). Hence the set \(E^4_k:={\mathcal {R}}_k(E^3_k)\) must satisfy \(P(E^4_k;\partial \Omega )=0\). By Lemma A.3, we can choose \(\varepsilon _k>0\) sufficiently small in order to ensure that \(|E^4_k\bigtriangleup E^3_k|<\frac{1}{k}\). Now choose \(\eta _k\in \left( 0,\frac{1}{2k}\right) \) such that \(P(E^3_k;Q_k)\le 2P(E^3_k;\partial \Omega )\), where
Since \(\Omega \) is bounded, possibly choosing \(\varepsilon _k>0\) even smaller, we can also ensure that \(\Omega \bigtriangleup {\mathcal {R}}^{-1}(\Omega )\subset Q_k\). Hence we can estimate
We can thus set \(E_k:=E^4_k\) for all \(k\in {\mathbb {N}}\) and the proof is complete. \(\square \)
Remark A.5
(A minor gap in the proof of [3, Proposition 15]) We warn the reader that the cut-off and the extension steps presented above were not mentioned in the proof of [3, Proposition 15], although they are unavoidable for the correct implementation of the rotation argument in the last step. Indeed, in general, one cannot expect the existence of a rotation \({\mathcal {R}}\in \mathrm {SO}(n)\) arbitrarily close to the identity map such that \(P({\mathcal {R}}(E);\partial \Omega )=0\) and, at the same time, the difference between \(P({\mathcal {R}}(E);\Omega )\) and \(P(E;\Omega )\) is small. For example, one can consider \(n=2\),
and
where \(A=\{(x_1,x_2)\in {\mathbb {R}}^2 : x_1>0,\ x_2>0\}\). In this case, for any rotation \({\mathcal {R}}\in \mathrm {SO}(2)\) arbitrarily close to the identity map, we have \(P({\mathcal {R}}(E);\Omega )>2+P(E;\Omega )\).
We conclude this section with the following result, establishing an approximation of \(BV_{{{\,\mathrm{loc}\,}}}\) functions similar to that given in Theorem A.4.
Theorem A.6
Let \(\Omega \subset {\mathbb {R}}^n\) be a bounded open set with Lipschitz boundary and let \(f\in BV_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\). There exists \((f_k)_{k\in {\mathbb {N}}}\subset BV({\mathbb {R}}^n)\) such that
for all \(k\in {\mathbb {N}}\) and
as \(k\rightarrow +\infty \). If, in addition, \(f\in L^1({\mathbb {R}}^n)\), then \(f_k\rightarrow f\) in \(L^1({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \).
Proof
We argue as in the proof of Theorem A.4, in two steps.
Step 1: cut-off at infinity. Since \(\Omega \) is bounded, we find \(R_0>0\) such that \({\overline{\Omega }}\subset B_{R_0}\). Given \((R_k)_k\subset (R_0,+\infty )\), we set \(g_k:=f\chi _{B_{R_k}}\) for all \(k\in {\mathbb {N}}\). By Theorem A.1, we have \(g_k\in BV({\mathbb {R}}^n)\) for a suitable choice of the sequence \((R_k)_{k\in {\mathbb {N}}}\), with \(|Dg_k|(\Omega )=|Df|(\Omega )\) for all \(k\in {\mathbb {N}}\) and \(g_k\rightarrow f\) in \(L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \). If, in addition, \(f\in L^1({\mathbb {R}}^n)\), then \(g_k\rightarrow f\) in \(L^1({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \).
Step 2: extension and cut-off near \(\Omega \). Let us define
for all \(k\in {\mathbb {N}}\). Since \(g_k\chi _\Omega \in BV(\Omega )\) with \(|Dg_k|(\Omega )=|Df|(\Omega )\) for all \(k\in {\mathbb {N}}\), by [4, Definition 3.20 and Proposition 3.21] there exists a sequence \((h_k)_{k\in {\mathbb {N}}}\subset BV({\mathbb {R}}^n)\) such that
for all \(k\in {\mathbb {N}}\) and
(the latter property easily follows from the construction performed in the proof of [4, Proposition 3.21] Now let \((v_k)_{k\in {\mathbb {N}}}\subset C^\infty _c({\mathbb {R}}^n)\) be such that \({{\,\mathrm{supp}\,}}v_k\subset A^c_k\) and \(0\le v_k\le 1\) for all \(k\in {\mathbb {N}}\) and \(v_k\rightarrow \chi _{ {}\overline{\Omega }^c}\) pointwise in \({\mathbb {R}}^n\) as \(k\rightarrow +\infty \). We can thus set \(f_k:=h_k+v_kg_k\) for all \(k\in {\mathbb {N}}\). By [4, Propositon 3.2(b)], we have \(v_kg_k\in BV({\mathbb {R}}^n)\) for all \(k\in {\mathbb {N}}\), so that \(f_k\in BV({\mathbb {R}}^n)\) for all \(k\in {\mathbb {N}}\). Since we can estimate
for all \(k\in {\mathbb {N}}\), we have \(f_k\rightarrow f\) in \(L^1_{{{\,\mathrm{loc}\,}}}({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \) (because \(\partial \Omega \) is Lipschitz, so \({\mathscr {L}}^{n}(\partial \Omega ) = 0\)), with \(f_k\rightarrow f\) in \(L^1({\mathbb {R}}^n)\) as \(k\rightarrow +\infty \) if \(f\in L^1({\mathbb {R}}^n)\). By construction, we also have
for all \(k\in {\mathbb {N}}\). The proof is complete.\(\square \)
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Comi, G.E., Stefani, G. A distributional approach to fractional Sobolev spaces and fractional variation: asymptotics I. Rev Mat Complut 36, 491–569 (2023). https://doi.org/10.1007/s13163-022-00429-y
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DOI: https://doi.org/10.1007/s13163-022-00429-y
Keywords
- Function with bounded fractional variation
- Fractional perimeter
- Fractional derivative
- Fractional gradient
- Fractional divergence
- Gamma-convergence