A remake of Bourgain–Brezis–Mironescu characterization of Sobolev spaces

We introduce a large class of concentrated p-Lévy integrable functions approximating the unity, which serves as the core tool from which we provide a nonlocal characterization of the Sobolev spaces and the space of functions of bounded variation via nonlocal energies forms. It turns out that this nonlocal characterization is a necessary and sufficient criterion to define Sobolev spaces on domains satisfying the extension property. We also examine the general case where the extension property does not necessarily hold. In the latter case we establish weak convergence of the nonlocal Radon measures involved to the local Radon measures induced by the distributional gradient.


Introduction
Let be an open subset of R d , d ≥ 1 and 1 ≤ p < ∞. We aim to provide a nonlocal characterization of first order Sobolev spaces on using the following type nonlocal energy forms where, ν : R d \{0} → [0, ∞) is measurable and satisfies the p-Lévy integrability condition

2)
This article is part of the section to "Theory of PDEs" editor by "Eduardo Teixeira".
B Guy Fabrice Foghem Gounoue guy.foghem@tu-dresden.de and a 1 (x, y) = min(1 (x), 1 (y)), a 2 (x, y) = max(1 (x), 1 (y)) and a 3 (x, y) = 1 2 (1 (x) + 1 (y)); where 1 is the indicator function of . Here and in what follows, the notation a ∧ b stands for min(a, b), a, b ∈ R. More explicitly, we can write the forms E i , as follows The nonlocal forms E i are crucial in the study of Integro-Differential Equations (IDEs) involving nonlocal operators of p-Lévy types; see for instance the recent works [12,16,18]. For p = 2, (1.2) is the well-known Lévy integrability condition. Actually, when ν is radial, the p-Lévy integrability (1.2) condition is consistent and self-generated in the sense that condition (1.2) holds true if and only if E 1 R d (u) < ∞ for all u ∈ C ∞ c (R d ); see Sect. 2.1 for the details. In addition, the p-Lévy integrability condition (1.2) indicates that ν is allowed to have a heavy singularity at the origin. For instance, ν(h) = |h| −d−sp satisfies the condition (1.2) if and only if s ∈ (0, 1).
Then u ∈ W 1, p ( ) for 1 < p < ∞ and u ∈ BV ( ) for p = 1. Moreover, there hold the estimates The constant K d,1 appearing in (1.7) is a universal constant independent of the geometry of and is given by the following general mean value formula over the unit sphere for any unit vector e ∈ S d−1 ; see Proposition 3.10 for the computation. The constant K d, p also appears in [5]. There is a similar constant in [22,Section 7] when studying nonlocal approximations of the p-Laplacian. Observe that in general, for every z ∈ R d , we have Theorem 1.1 yields the following nonlocal characterization of constant functions; see also [7].
Let us now comment about Theorem 1.1. Observing that, E 1 (u) ≤ E i (u), i = 1, 2, 3, Theorem 1.1 obviously remains true if the nonlocal forms of type E 1 are replaced with those of type E 2 or E 3 . It is to be noted that, Theorem 1.1 is governed by two fundamental counter intuitive remarks. Firstly, the lack of reflexivity of L 1 ( ) implies that, in the case p = 1, the function belongs to BV ( ) and not necessarily in W 1,1 ( ). In other words, assuming A 1 < ∞ is not enough to conclude that u ∈ W 1,1 ( ). We give here a mere counterexample in one dimension; see Counterexample 1 for the general case. For d = 1 and p ≥ 1 we consider, (1.10) For p = 1, it is straightforwards to verify that u ∈ BV (−1, 1)\W 1,1 (−1, 1) whereas we find that A 1 = 1. The second remark indicates that, the converse of Theorem 1.1 is not necessarily true in general. Indeed, by adopting the above example in (1.10), see also the counterexample 1, we find that u ∈ W 1, p ( ) while A p = ∞ for p > 1. A reasonable explanation to the latter matter is that, = (−1, 0) ∪ (0, 1) is not an extension W 1, p -domain. To put it another way, this situation in particular (and in general) occurs due to the lack of the regularity of the boundary ∂ . Therefore, to investigate the converse of Theorem 1.1, we need some additional assumption on such as the extension property. It is noteworthy to recall that ⊂ R d is called to be a W 1, p -extension (resp. a BV -extension) domain if there exists a linear operator E : It is to be noted that, in contrast to (1.11), having |∂ | = 0 does not necessarily imply (1.12). Indeed, it suffices to consider once more the example (1.10) where one gets ∇u = δ 0 (the Dirac measure at the origin), so that |∇u|(∂ ) = 1. Some authors rather define a BV -extension domain together with the condition (1.12); see for instance [1,17]. Extended discussions on BV -extension domains can be found in [23,24]. Several references on extension domains for Sobolev spaces can be found in [35]. Our second main result, which is an improved converse of Theorem (1.1), reads as follows. (1.13)

Moreover if p = 1 and is a BV -extension domain then for u
(1.14) We highlight that the counterexample 1 shows that the conclusion of Theorem In contrast to the forms of type E 1 , the collapse phenomenon across ∂ occurs for the forms of type E 2 or E 3 in Theorem 1.3.
Proof In fact, in both cases (i) and (ii) we have |∂ | = |∂ | = 0. Thus, Theorem 3.5 yields the first limit. For the case (ii), the remaining limits follow from Theorem 1.3 since  Note in particular that, Theorem 1.6 implies that (μ ε ) ε vaguely convergence to μ, i.e.
Let us comment on Theorem 1.  (1.4). Beside this, with the same assumptions, Bourgain-Brezis-Mironescu in [5,Theorem 2] also established the relation (1.13). The case = R d is also investigated by Brezis [7] while characterizing constant functions. The case p = 1, i.e., the relation (1.14), is also a natural subject of discussions in [5] wherein, the authors succeeded in the one dimensional setting when = (0, 1), viz., they proved that The general case d ≥ 2 was completed later in [9] when is a bounded Lipschitz domain. In this perspective, [9, Lemma 2] also established a variant of Theorem 1.6 for the case p = 1. Clearly, our setting of Theorem 1.1 is more general as no restriction on is required and, in the sense mentioned above, the class (ν ε ) ε satisfying (1.3) is strictly lager than that of (ρ ε ) ε satisfying (1.4). In addition, in contrast to [5], is not necessarily bounded in Theorem 1.3 and that the situation where has a Lipschitz boundary appears as a particular case of Theorem 1.3. We point out that Theorem 1.3 is reminiscent of [19,Theorem 3.4] for p = 2. Ultimately, let us mention that, after the release of the first version of this work, the authors of [3] brought to our attention that they also established the relation (1.13) when ν ε (h) = ε|h| −d−(1−ε) p (fractional kernels) for 1 < p < ∞. The case p = 1 is, however, not fully covered therein. Our approach in this paper extends the works from [5,7,9,28]. In the wake of [5], several works regarding the characterization of Sobolev spaces and alike spaces have emerged in the recent years. For example [27,29] for characterization of Sobolev spaces via families of anisotropic interacting kernels, [26,30] for characterization of BV spaces, [33] for a study of asymptotic sharp fractional Sobolev inequality, [6] for characterization of Besov type spaces of higher order and [19] for the study of Mosco convergence of nonlocal quadratic forms. This article is organized as follows. In the second section we address some examples of approximating sequence (ν ε ) ε and some nonlocal spaces in connection with function of type ν ε . The third section is devoted to the proofs of Theorem 1.1, Theorem 1.3 and Theorem 1.6.
Throughout this article, ε > 0 is a small quantity tending to 0. We frequently use the convex inequality (a + b) p ≤ 2 p−1 (a p + b p ) for a > 0, b > 0, the Euclidean scalar product of x = (x 1 , x 2 , · · · , x d ) ∈ R d and y = (y 1 , y 2 , · · · , y d ) ∈ R d is x ·y = x 1 x 1 +x 2 y 2 +· · ·+x d y d and denote the norm of x by |x| = √ x · x. The conjugate of p ∈ [1, ∞) is denoted by p , i.e. p + p = pp with the convention 1 = ∞. Throughout, |S d−1 | denotes the area of the d − 1-dimensional unit sphere, where we adopt the convention that |S d−1 | = 2 if d = 1.

p-Lévy integrability and approximation of Dirac measure
if ν({0}) = 0 and it satisfies the p-Lévy integrability condition; that is to say that

is called a Dirac approximation of p-Lévy measures.
Patently, one recovers the usual definition of Lévy measures when p = 2. Such measures are paramount in the study of stochastic process of Lévy type; see for instance [2,4,32] for further details. We intentionally omit the dependence of ν and ν ε on p. This dependence will be always clear from the context. The following result shows that by rescaling appropriately a radial p-Lévy integrable function ν(h) one obtains a family (ν ε ) ε satisfying (1.3).
We omit the remaining details as it solely involves straightforward computations.
The behavior of the rescaled family (ν ε ) ε in (2.1) when p = 2 is governed by two keys observations. The first is that it gives rise to a family of Lévy measures with a concentration property at the origin. Secondly, from a probabilistic point of view one obtains a family of pure jumps Lévy processes (X ε ) ε each associated with the measure ν ε (h)dh from a Lévy process X associated with ν(h)dh. In fact, the family of stochastic processes (X ε ) ε converges in finite dimensional distributional sense (see [19]) to a Brownian motion provided that one in addition assumes that ν is radial. Proposition 2.5 (ii) below shows that the generator of the process X ε denoted L ε (see (2.4)), converges to − 1 2d which is the generator of a Brownian motion. In short, rescaling via (2.1) any isotropic pure jump Lévy process leads to a Brownian motion. This could be one more argument to back up the ubiquity of the Brownian motion. The convergence highlighted above is involved in a more significant context. For example in [19], the convergence in Mosco sense of the Dirichlet forms associated with process in play is established. Beside these observations, the works [16,18] establish that if is bounded with a Lipschitz boundary and u ε satisfies in the weak sense nonlocal problems of the L ε u ε = f in augmented with Dirichlet condition u ε = 0 on c (resp. Neumann condition N u ε = 0 on c ) condition then (u ε ) ε converges in L 2 ( ) to some u ∈ W 1,2 ( ), where u is the weak solution to the local problem − 1 2d u = f in augmented with Dirichlet boundary u = 0 on ∂ (resp. Neumann condition ∂u ∂n = 0 on ∂ ). Here, L ε is given by (2.4) and N ε is defined by Thus for β > p we have In either case, letting δ → 0 provides the claim.

Remark 2.4
Assume the family (ν ε ) ε satisfies (1.3). Note that the relation is often known as the concentration property and is merely equivalent to lim ε→0| h|>δ Indeed, for all δ > 0 we havê Consequently, for all δ > 0 we also have The next result infers certain some convergences of the family (ν ε ) ε for the case p = 1 and p = 2.
where is the Laplace operator and L ε is the integrodifferential operator using the fundamental theorem of calculus we can write The conclusion clearly follows since | and by Remark 2.3 we have (ii) Note that D 2 u is bounded in a neighborhood of x. Hence, for 0 < δ < 1 sufficiently small, for all |h| < δ we have the estimate The boundedness of u implies that lim ε→0| h|>δ Since the Hessian of u is continuous at x, given η > 0 we have Thus, the leftmost expression vanishes since η > 0 is arbitrarily. Next, by symmetry we have´| h|≤δ h i h j ν ε (h) dh = 0 for i = j. The rotation invariance of the Lebesgue measure implieŝ Finally, by the fundamental theorem of calculus we find that Let us give examples of ν ε satisfying (1.3). The first example is related to fractional Sobolev spaces.

Example 2.6
The family (ν ε ) ε of kernels defined for h = 0 by The next class of examples is that of Proposition 2.2.
is radial and consider the family (ν ε ) ε such that each ν ε is the rescaling of ν defined as in (2.1) provided that A subclass is obtained if one considers an integrable radial function ρ : for which the p-Lévy integrability condition in (1.3) holds. Note that c ε → 1 as ε → 0.
Some special cases are obtained for β ∈ {0, ε p − d, p}. For the limiting case β = −d, we put For the limiting case β = −d consider In either case the constant b ε → 1 as ε → 0 and is such that´Rd Another example familiar to the case β = −d is s ∈ (0, 1), one recovers the Sobolev space of fractional order denoted W s, p ( ); see [10,18] for more. If ν has full support, the space W [15,18,19] for recent results involving this types of spaces. We recall that, 1 2

Local and nonlocal spaces
is the core energy space for a large class of nonlocal problems with Dirichlet, Neumann or Robin boundary conditions. See for instance [11,12,14,16,31]. If ⊂ R d has a sufficiently regular boundary or = R d then according to Theorem 1.3 and Theorem 1.5, it is legitimate to say that the nonlocal spaces W Let us recall the following standard approximation result for the space BV ( ); see [13, p. 172], [25,Theorem 14.9] or [1, Theorem 3.9]. Recall that, if a function u ∈ L 1 ( ) is regular enough, say, u ∈ W 1,1 ( ) then we have u ∈ BV ( ). From this we find that BV ( ) ∩ C ∞ ( ) = W 1,1 ( ) ∩ C ∞ ( ). Next, we establish some useful estimates. Note that for h ∈ R d we havê .
Therefore, for every u ∈ W 1, p (R d ) and h ∈ R d we havê .
(2.5) By Theorem 2.11 the BV -norm of an element in BV (R d ) can be approximated by the W 1,1norms of elements in W 1,1 (R d ). Whence for p = 1, (2.5) implies that, for u ∈ BV (R d ) and 1∧|h|) , for all u ∈ BV ( )).
Likewise, if p = 1 and u ∈ BV ( ) one gets the other estimate from the estimate (2.6).
An immediate consequence of Lemma 2.12 is the following embedding result. It is worth emphasizing that the above embeddings may fail if is not an extension domain (see the counterexample 1). Another straightforward consequence of Lemma 2.12 is the following.
If p = 1 and is a BV -extension domain we also have, The next proposition shows that the p-Lévy integrability condition is consistent and optimal in the sense that it draws a borderline for which a space of type W p ν ( ) is trivial or not.
be a compact set. Since |∇u| ∈ L p (K ), for arbitrary η > 0 there is The choice 0 < δ < dist(K , ∂ ) ensures that B δ (x) ⊂ for all x ∈ K . From the foregoing, using the fundamental theorem of calculus, polar coordinates and the formula (1.9) yield Therefore, for each η > 0 and each compact set K ⊂ we have Since u W p ν ( ) < ∞ and C δ = ∞, this is possible only if ∇u p L p (K ) = 0. As the compact set K ⊂ is arbitrary, we find that ∇u = 0 a.e. on . Thus u is a constant since is connected. (iv) The upper inequality clearly follows from (2.5). Proceeding as for the estimate (2.8) by taking = R d and K = R d also yields that, for all η > 0 there is δ = δ(η) > 0 such that . This estimate remains true for any δ > 0, if ∇u L p (R d ) = 0.
The next theorem provides a characterization of the p-Lévy integrability condition.

This remains true when
The right hand side of the estimate (2.7) implies the continuity of the embedding and hence C δ =´B δ (0) |h| p ν(h) dh < ∞. Next, we fix n ≥ 1 such that δ > 2 n so that supp u n ⊂ B δ/2 (0).

Main results
First and foremost, the proof of Theorem 1.3 in the case = R d is much simpler. Indeed, by the estimates (2.9) and (3.10) below, for sufficiently small η > 0, there is δ = δ(η) > 0 such that Letting ε → 0 and η → 0 successively, using the formulas (2.3) and (2.2), we get . (3.1) The case p = 1 and u ∈ BV (R d ) can be proved analogously. In fact, it can be shown that (3.1) holds if and only if up to a multiple factor (ν ε ) ε satisfies (1.3). In other words, the class (ν ε ) ε is the largest (the sharpest) class for which the BBM formula (3.1) holds. From now on, we assume = R d . We start with the following lemma which is somewhat a revisited version of [5, Lemma 1].

Lemma 3.1 Assume
Proof Let us introduce the truncated measure ν δ (h) = |h| −1 (1 ∧ |h| p )ν(h) 1 R d \B δ (h) for δ > 0 which enables us to rule out an eventual singularity of ν at the origin. Moreover, note that ν δ ∈ L 1 (R d ). It turns out that the mappings (x, y) → u(x)ϕ(y) ν δ (x − y) and (x, y) → u(x)ϕ(x) ν δ (x − y) are integrable. Indeed, using Hölder inequality combined with Fubini's theorem yield Analogously, we also geẗ Consequently, by interchanging x and y, using Fubini's theorem and the symmetry of ν we obtain( Thus letting δ → 0 implies ( Likewise one has ( Summing the estimates (3.2) and (3.3) gives the desired inequality. (1.6). Then given a unit vector e ∈ S d−1 and ϕ ∈ C ∞ c (R d ) with support in the following estimate holds true

Theorem 3.2 Let ⊂ R d be an open, u ∈ L p ( ), p ≥ 1 and A p be defined as in
Proof Throughout, to alleviate the notation we denote π ε (x − y) First, for δ = dist(supp(ϕ), ∂ ) > 0, the Hölder inequality implieŝ Second, using again the Hölder inequality and |h| − p (1 ∧ |h| p ) ≤ 1 we find thaẗ Therefore inserting these two estimates in the previous identity and combining the resulting estimate with that of Lemma 3.1 imply (3.5) It remains to compute the limits appearing on the left hand side of (3.5). For all x, h ∈ R d we have Thus, using the above expression and the fact that´Rd Let (e, v 2 , · · · v d ) be an orthonormal basis of R d in which we write the coordinates w = (w 1 , w 2 , · · · , w d ) = (w 1 , w ) that is w 1 = w · e and w i = w · v i . Similarly, in this basis one has ∇ϕ(x) = (∇ϕ(x) · e, (∇ϕ(x)) ). Observe that ∇ϕ(x) · w = ∇ϕ(x) · e (w · e) + [∇φ(x)] · w . We find that Consider the rotation O(w) = (w 1 , −w ) = (w · e, −w ) then the rotation invariance of the Lebesgue measure entails that dσ d−1 (w) = dσ (O(w)) and we havê Whereas, by symmetry we havê Altogether, we find that In conclusion, Analogously one is able to show that By substituting the two relations (3.6) and (3.7) in (3.5) and using the dominate convergence theorem one readily ends up with the desired estimate.

Proof of Theorem 1.1 The estimate (3.4) holds true for all
In virtue of the density of C ∞ c ( ) in L p ( ), it readily follows from (3.4) that for each i = 1, · · · , d the mapping ϕ →´ u(x)∂ x i ϕ(x) dx uniquely extends as a continuous linear form on L p ( ). Since 1 < p < ∞, the Riesz representation for Lebesgue spaces reveals that there exists a unique g i ∈ L p ( ) and we set ∂ x i u = −g i , such that In order words, u ∈ W 1, p ( ). Further, the L p -duality and (3.4) yields the estimate (1.7) as follows Hence u ∈ BV ( ) and we have |u| BV ( ) ≤ d A 1 K d,1 which is the estimate (1.7).
The next result improves the estimate (1.7).
Moreover if p = 1 and u ∈ L 1 ( ) then we have We assume that u is extended by zero off and let u δ = u * φ δ is the convolution product of u and φ δ . If z ∈ δ and |h| ≤ δ then z − h ∈ δ − h ⊂ . A change of variables implies Thus given that´φ δ dh = 1, integrating with respect to φ δ (h)dh, Jensen's inequality yields¨ In other words, we havë (3.8) Note that u δ ∈ C ∞ (R d ) and δ, j = δ ∩ B j (0) has a compact closure for each j ≥ 1. Then for each j ≥ 1 the Lemma 3.6 implies Tending j → ∞ in the latter we get The only interesting scenario occurs if A p < ∞. In this case, Theorem 1.1 ensures that u ∈ W 1, p ( ). Clearly we have ∇u δ = ∇(u * φ δ ) = ∇u * φ δ and φ δ * ∇u − ∇u L p ( ) → 0 as δ → 0. The desired inequality follows by letting δ → 0 in (3.9) since Case p = 1: Again we only need to assume that A 1 < ∞ so that by Theorem 1.1, u ∈ BV ( ). The relation (3.9) implies that Thus, since u is a distribution on we get This completes the proof since the above holds for arbitrarily chosen χ ∈ C ∞ c ( , R d ) such that χ L ∞ ( ,R d ) ≤ 1, by definition of | · | BV ( ) and the previous estimate we get

Second proof.
Here is an alternative. Since for all δ > 0,´B δ (0) |h| p ν ε (h) dh → 1 as ε → 0 (see the formula (2.3)), for each compact set K ⊂ and η > 0 inequality (2.8) implies Let K j = j ⊂ j+1 and ( j ) j be an exhaustion of . Since the above inequality is true for every compact set K = K j ⊂ and every η > 0 we conclude that The case p = 1 and u ∈ BV ( ) follows from the approximation Theorem 2.11.
It is worth to mention that the convolution technique used in the first proof above was first used in [7] when = R d and also appears in [28]. The next theorem is a the counterpart of Theorem 3.3 and is a refinement version of Theorem 2.14.

Theorem 3.4 Let ⊂ R d be a W 1, p -extension domain and u
Moreover, for p = 1, if is a BV -extension domain and u ∈ L 1 ( ) then we have Proof The cases ∇u L p ( ) = ∞ and |u| BV ( ) = ∞ are trivial. Assume u ∈ W 1, p ( ) and let u ∈ W 1, p (R d ) be its extension to R d . Consider (δ) = + B δ (0) = {x ∈ R d : dist(x, ) < δ} be a neighborhood of where 0 < δ < 1. We claim that for each ε > 0, the following estimate holds (3.10) For each n ≥ 1, passing through the polar coordinates and using the identity (1.9) we find thaẗ The estimate (3.10) clearly follows since we havê ˆ Letting ε → 0 the relation (3.10) yields Recalling that u ∈ W 1, p (R d ), u = u | and using (1.11) the desired estimate followŝ The estimate (3.10) applied to u n and the Fatou's lemma yield Correspondingly, we also get the estimatë Therefore, letting ε → 0 implies that Recalling that u ∈ BV (R d ), u = u | and ∂ satisfies (1.12), i.e., |∇u|(∂ ) = 0 we have The following result involves the collapse across the boundary ∂ .

Theorem 3.5 Assume ⊂ R d is open then for any u
Accordingly, together with (3.1), we deduce the desired result as follows The reverse inequality follows analogously, since by exploiting (3.10) (or (3.11)) one easily gets Next we establish a pointwise and L 1 ( ) convergence when u is a sufficiently smooth function.

Lemma 3.6 Let ⊂ R d be open and u
The following convergence occurs in both pointwise and L 1 ( ) sense: Proof First proof of Lemma 3.6. Let σ > 0 be sufficiently small. By assumption ∇u is uniformly continuous and hence one can find 0 < η = η(σ ) < 1 such that if |x − y| < η then |∇u(y) − ∇u(x)| ≤ σ. (3.12) Let In virtue of the fundamental theorem of calculus, we have The mapping s → G p (s) = |s| p belongs to C 1 (R d \{0}) and G p (s) = pG p (s)s −1 . Thus, we have + s(b − a)) ds.
Set a = ∇u(x) · h and b =´1 0 ∇u(x + th) · h dt so that the relation (3.12) yields Integrating both sides with respect to ν ε (h) dh, implies that Since´| h|≤η x |h| p ν ε (h) dh → 1 as ε → 0, by the formula (2.3), letting ε → 0 and σ → 0 successively yields R(x, ε) → 0. Whereas, using polar coordinates, the relation (1.9) and the Remark 2.3 giveŝ Furthermore, a close look to our reasoning reveals that we have subsequently shown that This is due to the fact that, for all δ > 0 we havê Therefore, letting ε → 0 in the latter expression and taking into account Remark 2.3 gives The pointwise convergence (3.15) readily follows. Since on the other side, we havê Thus the remaining details follow by proceeding as in the previous proof.
We are now in position to prove Theorem 1.3.
Proof of Theorem 1.3 Assume A p = ∞ then by Theorem 1.1 we have ∇u L p ( ) = ∞ for 1 < p < ∞ and |u| BV ( ) = ∞ for p = 1. In either case the relation (1.13) or (1.14) is verified. The interesting situation is when A p < ∞, i.e., by Theorem 1.1, u ∈ W 1, p ( ) if 1 < p < ∞ and u ∈ BV ( ) if p = 1. We provide two alternative proofs. As first alternative, the result immediately follows by combining Theorem 3.3 and Theorem 3.4. For the second alternative, consider 1 < p < ∞ or u ∈ W 1,1 ( ). By Lemma 2.12 there is C > 0 independent of ε such that for u, v ∈ W 1, p ( ), Therefore, it suffices to establish the result for u in a dense subset of W 1, p ( ). Note that We conclude by using Lemma 3.6.
As consequence of Theorem 1.3 we have the following concrete examples.

Corollary 3.7 Assume
⊂ R d is an extension domain and u ∈ L p ( ). If we abuse the notation ∇u L 1 ( ) = |u| BV ( ) for p = 1, then there holds Proof For the first relation, take For the second and third take ν ε (h) = d+β Using (3.10) and (3.11) with replaced by E implyÊ Hence, since´| h|>δ ν ε (h) dh ≤ 1, the family of functions (μ ε ) ε is bounded in L 1 (E). In virtue of the weak compactness of L 1 (E) (see [8, p. 116]) we may assume that (μ ε ) ε converges in the weak-* sense to a Radon measure μ E , i.e., μ ε − μ E , ϕ ε→0 −−→ 0 for all ϕ ∈ C(E) otherwise, one may pick a converging subsequence. For a suitable ( j ) j∈N exhaustion of , i.e., j s are open, each K j = j is compact, K j = j ⊂ j+1 and = j∈N j , it is sufficient to let μ = μ K j = K d, p |∇u| p on K j . We aim to show that μ = K d, p |∇u| p . Noticing μ and K d, p |∇u| p are Radon measures it sufficient to show that both measures coincide on compact sets, i.e., we have to show that μ E (E) = K d, p´E d|∇u| p (x). On the one hand, since μ ε (E) → μ(E) and´| h|>δ ν ε (h) dh → 0 as ε → 0, the fact that u ∈ W 1, p ( ) or u ∈ BV ( ) enables us to successively let ε → 0 and δ → 0 in (3.16) which amounts to the followingÊ On other hand, since E has a nonempty interior, Theorem 3.3 implies Finally μ(E) = μ E (E) = K d, p´E d|∇u| p (x). Whence we get dμ = K d, p d|∇u| p as claimed.
A consequence of Theorem 1.6 is given by the following analog result. Next, we sate without proof the asymptotically compactness involving the case where the function u also varies. The full proof can be found in [18,Theorem 5.40] and [28]. There exists a subsequence (ε n ) n with ε n → 0 + as n → ∞ such that (u ε n ) n converges in L p ( ) to a function u ∈ L p ( ). Moreover, u ∈ W 1, p ( ) if 1 < p < ∞ or u ∈ BV ( ) if p = 1. (i) Clearly, u ∈ W 1, p ( ) for all 1 ≤ p < ∞ with ∇u = 0 on . Note however that, the weak derivative of u on (−1, 1) is δ 0 ; the Dirac mass at the origin. It follows that u / ∈ W 1, p (−1, 1) for all 1 ≤ p < ∞ and u ∈ BV (−1, 1) with |u| BV (−1,1) = 1.
(ii) Moreover, is not a W 1, p -extension domain. Indeed, assume u ∈ W 1, p (R) is an extension of u defined. In particular, u ∈ W 1, p (−1, 1) and u = u on . The distributional derivative of u on (−1, 1) is ∇u = δ 0 , This contradicts the fact that u ∈ W 1, p (R). (iii) Since integrals disregard null sets, we have u W s, p ( ) = u W s, p (−1,1) for all 1 ≤ p < ∞. If 1 < p < ∞ and s ≥ 1/ p then u W s, p ( ) = u W s, p (−1,1) = ∞ and hence u / ∈ W s, p ( ). Thus the embedding W 1, p ( ) → W s, p ( ) fails. However, if p = 1 we get u ∈ W s,1 (−1, 1). Since s = 1 − ε this also implies that Case d ≥ 2. The above example persists in higher dimension. Consider be the unit ball B 1 (0) deprived with the hyperplane {x d = 0} that is, Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article's Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article's Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.