Representation formulas for pairings between divergence-measure fields and $BV$ functions

The purpose of this paper is to find pointwise representation formulas for the density of the pairing between divergence-measure fields and BV functions, in this way continuing the research started in [17,20]. In particular, we extend a representation formula from an unpublished paper of Anzellotti [7] involving the limit of cylindrical averages for normal traces, and we exploit a result of [35] in order to derive another representation in terms of limits of averages in half balls.

The current general setting for the pairing theory is the following (see e.g.[6,11,15]).Given an open set Ω ⊂ R N , we say that a vector field A ∈ L ∞ (Ω; R N ) is a divergencemeasure field, and we write A ∈ DM ∞ (Ω), if div A is a finite Radon measure on Ω.For any function u ∈ BV (Ω) ∩ L ∞ (Ω), the pairing (A, Du) is defined in the sense of distributions as We recall that this definition is well posed, since the measure div A does not charge sets of (N − 1)-dimensional Hausdorff measure zero, while the precise representative u * of a BV function u is defined H N −1 -a.e. in Ω.In fact, it is proved in [6,11] that the pairing (A, Du) is a Radon measure in Ω, and | (A, Du) | ≤ A L ∞ (Ω;R N ) |Du|, so that there exists a density θ(A, Du, •) ∈ L 1 (Ω, |Du|) such that the equality (A, Du) = θ(A, Du, x)|Du| holds in the sense of measures.In addition, in [17,20,21] it is shown that the boundedness assumption on u can be replaced with the weaker requirement that u * ∈ L 1 (Ω; | div A|).
The aim of this paper is to find some representation formula for the pairing, obtained through a detailed description of the density θ(A, Du, x) of the pairing measure (A, Du) with respect to |Du|.
In the classical case, when A ∈ DM ∞ (Ω) ∩ C(Ω; R N ) and u ∈ BV (Ω) ∩ C 1 (Ω), then (A, Du) = A • ∇u L N , where L N is the N -dimensional Lebesgue measure, so that θ(A, Du, x) = 0 if ∇u(x) = 0, whereas The absolutely continuous part satisfies (A, Du) a = A • ∇u L N , as it is first shown in [11], while in [17] the authors prove that (A, Du) j = Tr * (A, J u ) |D j u|, where Tr * (A, J u ) is the average of the interior and exterior weak normal traces of the vector field A on the jump set J u of u (see Section 2.3 below).This result is satisfactory for what concerns the representation of the absolutely continuous part and the jump part of the pairing measure, since it implies that θ(A, Du, x) = A • ∇u(x) |∇u(x)| χ {∇u =0} (x) , for L N -a.e.x ∈ Ω, and θ(A, Du, x) = Tr * (A, J u )(x) , for |D j u|-a.e.x ∈ Ω.
On the other hand, for what concerns the representation of the Cantor part of the pairing measure, in [17] it is proved that (A, Du) c (Ω \ S A ) = A • D c u (Ω \ S A ), where S A is the approximate discontinuity set of A, and A denotes the approximate continuous representative of A in Ω \ S A .Yet, this representation on Ω \ S A is far from being optimal, since there exist vector fields A ∈ DM ∞ (R N ) such that the Hausdorff dimension of S A is N (see Example 3.9).
Aiming to obtain a representation formula for θ(A, Du, •) without any additional assumption, we propose a new approach based on the use of the coarea formula for the pairing measure proved in [17,Theorem 4.2].The basic idea is to use the representation of the purely jump measure (A, Dχ {u>t} ) on superlevel sets of u, and to recover information on (A, Du) through the coarea formula, obtaining in particular that (A, Du) c = Tr * (A, ∂ * {u > u(•)})(•)|D c u| where u is the approximate limit of u at some Lebesgue point and ∂ * E denotes the reduced boundary of some measurable set E (see Theorem 3.12).
Hence, more explicit representation formulas for θ(A, Du, •) can be inherited by explicit representation formulas for the weak normal traces of A.
A relevant contribution in this direction is contained in the unpublished paper [7], where the divergence of the vector field A ∈ L ∞ (Ω; R N ) is assumed to be a summable function, and the weak normal trace of A is obtained as the limit of a suitable cylindrical average.More precisely, [7,Theorem 3.6] where Du = ν u |Du| is the polar decomposition of Du, and, for some set G ⊂ Ω and some whenever the limits exist, with (the existence of the limit in (1) for |Du|-a.e.x ∈ Ω is part of the statement).
In Theorem 4.10 we obtain a generalization of above-mentioned result to divergencemeasure vector fields, by adapting the arguments of the original proof through the use, as a new ingredient, of the Gauss-Green formulas recently proved in [17] (see Theorem 4.2).More precisely, the representation formula (1) turns out to hold true provided that the set on which the jump part of the measure div A is concentrated, which we denote by Θ A , has an H N −1 -negligible intersection with the jump set J u of the function u (see also Remark 4.11).This result is optimal: if instead H N −1 (Θ A ∩ J u ) > 0, then relation ( 1) is no longer valid, as it is shown in Example 4.1.
As an application, we obtain the following Gauss-Green formula, valid for every , and E ⊂ R N set of finite perimeter such that supp(χ E u) is bounded: where E 1 denotes the measure theoretic interior of E, and u i denotes the interior trace of u on ∂ * E (see Theorem 4.9).
In addition, in the paper [19] the cylindrical averages approach is further exploited in order to gain an explicit representation for the relaxation of a pairing-type functional.
Finally, combining our results with the representation formula for weak normal traces obtained in [35,Theorem 4.4], we get the following further integral representation for |Du|-a.e.x ∈ Ω, where As a further application of our general representation formula, in the final part of the paper we recover the local structure of the pairing measure by means of its tangent measures, coherently with the classical theory of sets of finite perimeter and functions of bounded variation.This result can be also achieved by means of direct calculations on the blow-up sequence of the pairing measure; however this is not necessary, since we can exploit Theorem 3.12 below and the Federer-Vol'pert theorem.
The plan of the paper is the following.In Section 2 we set the notation, and we recall some results on divergence-measure vector fields, their weak normal traces and functions of bounded variation.
In Section 3 we first recall some results concerning the pairing between divergencemeasure vector fields and functions of bounded variation, mainly taken from [6,11,17,20].Then we prove the result on the representation of the density θ(A, Du, •) in terms of weak normal traces (Theorem 3.12).
In Section 4, building on the results of Section 3, we show that θ(A, Du, •) can be represented in terms of the cylindrical averages introduced in [7]; then we achieve a similar result with the half balls averages introduced in [35].
Finally, in Section 5 we briefly describe the tangential properties of the pairing measure.

Notation and preliminary results
In the following we denote by Ω a nonempty open subset of R N , and for every set E ⊂ R N we denote by χ E its characteristic function.We say that a set E is compactly contained in Ω, and we write E ⋐ Ω, if the closure E of E is a compact subset of Ω.Given two sets E, F ⊂ R N , their symmetric difference is the set E △ F := (E \ F ) ∪ (F \ E).For x ∈ R N and r > 0, we denote by B r (x) the ball centered in x with radius r, and we set B 1 := B 1 (0).2.1.Measures.We denote by L N and H N −1 the Lebesgue measure and the (N − 1)dimensional Hausdorff measure in R N , respectively.Unless otherwise stated, a measurable set is a L N -measurable set.We set ω N := L N (B 1 ).
Following the notation of [3], we denote by M loc (Ω) the space of Radon measures on Ω, and by M(Ω) the space of finite Radon measures on Ω.
Given µ ∈ M(Ω) and a µ-measurable set E, the restriction µ E is the Radon measure defined by µ E(B) The total variation |µ| of µ ∈ M(Ω) is the nonnegative Radon measure defined by for every µ-measurable set E. Since µ ∈ M loc (Ω) if and only if µ ∈ M(Ω ′ ) for every open set Ω ′ ⋐ Ω, the above definitions can be easily extended to the case of a not necessarily finite Radon measure µ by adding the assumptions B ⋐ Ω and E h ⋐ Ω, respectively.
In this case, we say that x is a Lebesgue point of f with respect to µ. Thanks to Lebesgue's differentiation theorem, we know that µ-almost every x ∈ Ω is a Lebesgue point of f with respect to µ.In addition, in every Lebesgue point of f with respect to µ the approximate limit is uniquely determined and is denoted by In what follows we choose f as pointwise representative of f ∈ L 1 loc (Ω, µ); that is, we assume f (x) := f (x) in every Lebesgue point and whenever this choice does not cause any ambiguity.
2.2.Divergence-measure fields.We denote by DM ∞ (Ω) the space of all vector fields A ∈ L ∞ (Ω; R N ) whose divergence in the sense of distributions is a finite Radon measure in Ω, acting as Similarly, DM ∞ loc (Ω) will denote the space of all vector fields A ∈ L ∞ loc (Ω; R N ) whose divergence in the sense of distributions is a Radon measure in Ω.
The main property, proved in [11, Proposition 3.1] (see also [35,Theorem 3.2]), is that for every A ∈ DM ∞ loc (Ω) the measure div A is absolutely continuous with respect to the Hausdorff measure H N −1 , so that the following decomposition result holds, which is the localized version of [20, Proposition 2.3] (see also [2,Proposition 2.3]).Proposition 2.1.Given a vector field A ∈ DM ∞ loc (Ω), the set is a Borel set, σ-finite with respect to H N −1 , and the measure div A can be decomposed as the sum of mutually singular measures div A = div a A + div c A + div j A, where In what follows, we will call Θ A the jump set of the measure div A.

Weak normal traces on oriented countably H
A notion of orientation on rectifiable sets can be given as follows: if we choose oriented hypersurfaces (Σ k ), we define H N −1 -a.e. on Σ an orientation ν Σ by selecting pairwise disjoint Borel sets This orientation depends clearly on the choice of the decomposition, but only up to the sign, due to the fact that for any pair of In what follows, we will deal with the traces of the normal component of a vector field A ∈ DM ∞ loc (Ω) on an oriented countably H N −1 -rectifiable set Σ ⊂ Ω.In order to fix the notation, we briefly recall the construction given in [1, Propositions 3.2, 3.4 and Definition 3.3].Given a domain Ω ′ ⋐ Ω of class C 1 , the trace of the normal component of A on ∂Ω ′ is the distribution defined by ( 4) It turns out that this distribution is induced by an L ∞ function on ∂Ω ′ , still denoted by Tr(A, ∂Ω ′ ), and (5) Given an oriented countably H N −1 -rectifiable set Σ, and using the notation for the covering of Σ introduced at the beginning of this section, one can prove that for every k ∈ N, there exist two open bounded sets Ω k , Ω ′ k with C 1 boundary and interior normal vectors By a deep localization property proved in [1, Proposition 3.2], we can fix an orientation on Σ, given by ν and the interior and exterior normal traces of A on Σ are defined by As a consequence, if we consider two oriented countably H N −1 -rectifiable sets Σ and Σ ′ with the same orientation and such that Moreover, the normal traces belong to L ∞ (Σ, H N −1 Σ) and satisfy for every open set Ω ′ such that Σ ⊂ Ω ′ ⋐ Ω (see for instance [15,Theorem 4.2]), and In particular, by ( 5), we get In what follows we use the notation Remark 2.2.We stress the fact that in (4) we are using the opposite sign with respect to the definition of normal trace given in [1,6], and so the opposite orientation of the rectifiable hypersurfaces.Anyway, if Σ is oriented by a normal vector field ν and Σ ′ is the same set oriented by ν ′ := −ν, then so that the difference Tr i (A, Σ) − Tr e (A, Σ) in ( 8) is independent of the choice of the orientation on Σ.
2.4.Functions of bounded variation.Even if we mostly follow the terminology of [3], nevertheless we recall the main conventions and results for reader's convenience.
A function u ∈ L 1 (Ω) has bounded variation in Ω, and we write u ∈ BV (Ω), if the distributional derivative Du of u is a vector valued finite Radon measure in Ω.We denote by BV loc (Ω) the set of functions u ∈ L 1 loc (Ω) that belongs to BV (Ω ′ ) for every open set Ω ′ ⋐ Ω.In addition, we let BV (Ω; R m ) be the space of R m -vector valued functions of bounded variations in Ω, and we define analogously the local space BV loc (Ω; R m ).
In spite of the fact that a BV function u is an L 1 function, it admits a representative well defined outside an H N −1 -negligible set.In order to define it, we recall some more results on approximate limits of summable functions.
If in the definition of approximate limit (2) we have µ = L N and f = u ∈ L 1 loc (Ω; R m ), then we say that x is a Lebesgue point of u, omitting the reference to the Lebesgue measure.In order to emphasize the distinction with the approximate jump points of u defined below, we use here and in similar situations the notation u(x) for the pointwise representative of u in its Lebesgue points.The set C u ⊂ Ω of points where this property holds is called the approximate continuity set of u, whereas the set S u := Ω \ C u is called the approximate discontinuity set of u.
We say that x ∈ Ω is an approximate jump point of u if there exist a, b ∈ R m , a = b, and a unit vector ν ∈ R N such that and ( 9)  9) up to a permutation of (a, b) and a change of sign of ν, is denoted by (u i (x), u e (x), ν u (x)).The set of approximate jump points of u is denoted by J u , and it is clear that J u ⊂ S u .
If u ∈ BV loc (Ω; R m ), then both J u and S u are countably H N −1 -rectifiable sets, we have H N −1 (S u \ J u ) = 0, and for H N −1 -a.e.x ∈ J u the unit vector ν u (x) can be identified with the normal vector ν Ju (x) defined in Section 2.3 for general countably H N −1 -rectifiable sets (up to a change in orientation).
In the remaining part of this section, we focus on the scalar case m = 1.The gradient measure Du of a function u ∈ BV (Ω) can be decomposed as the sum of mutually singular measures Du = D a u + D j u + D c u, where D a u is the absolutely continuous part with respect to the Lebesgue measure, that is, D a u = ∇u L N (∇u ∈ L 1 (Ω; R N ) is the approximate differential of u), while D j u is the jump part, characterized by D j u = (u i − u e ) ν u H N −1 J u , and D c u is the Cantor part.We denote by Based on the notion of density of a measurable set E at a point x ∈ R N : (whenever the limit exists), we define the measure theoretic interior and exterior of E: as well as the measure theoretic boundary In the case of a general measurable function u : Ω → R, we set A measurable set E is of (locally) finite perimeter in Ω if its characteristic function χ E belongs to BV (Ω) (respectively, BV loc (Ω)).If E has locally finite perimeter in Ω, we call reduced boundary ∂ * E of E the set of all points x ∈ Ω in the support of |Dχ E | such that the limit where ν ∂ * E is the normal vector to ∂ * E, in the sense of Section 2.3.Due to these facts, with a little abuse of notation, we shall simply write ν E to denote the measure theoretic unit interior normal, coherently with most of the literature.
If u ∈ BV loc (Ω), then the level sets E t := {u > t} are of locally finite perimeter for L 1 -a.e.t ∈ R, and we have ν Et (x) = ν Σt (x) = ν u (x) for H N −1 -a.e.x ∈ Σ t , where Σ t := ∂ * {u > t}.In addition, the measure Du can be disintegrated on the level sets of u thanks to the coarea formula (see [24, Theorem 4.5.9]).Thanks to Theorem 2.4 and the inclusion we deduce that Specializing the coarea formula to the approximate continuity set C u , and using the inclusion we also get (13)

The pairing measure and its representation
In order to give the notion of pairing between divergence-measure fields and BV functions, we need a particular subset of the BV space, previously introduced in [20].
We remark that | div A| ≪ H N −1 and u * is defined H N −1 -a.e. in Ω, hence these definitions are well-posed.
We introduce now the general notion of pairing between a divergence-measure field and a suitable BV function (see [17, Section 2.5 and Theorem 4.12]).Definition 3.2 (Pairing).The pairing between a vector field A ∈ DM ∞ loc (Ω) and a func- , and obtain the same spaces.This observation allows the authors of [20] to give a more general definition of pairing involving, instead of u * , the λ-representative u λ given by ( 10): more precisely, they define the λ-pairing (A, Du) λ , acting as Since (A, Du) and (A, Du) λ differ only on Θ A ∩ J u , by [20,Proposition 4.4], we are going to state our results for the standard pairing (A, Du), underlining possible differences only whenever they appear. 1  The relevant properties of the pairings are recalled in the following proposition, which is the combination of [17,Theorem 4.12] and [21, Proposition 2.
In what follows we will write (i) absolutely continuous part: In addition, we denote by (A, Du) d the diffuse part of the pairing measure; that is, Remark 3.6.Proposition 3.4 can be seen as the particular case λ ≡ 1/2 of [20, Proposition 4.4], which applies to the general λ-pairing given by ( 14) and provides an estimate analogous to (15): given any Borel function λ : Ω → [0, 1], we have We point out that in [20] the authors denote by (A, Du) * the standard pairing (for λ ≡ 1/2), whereas we use the classical notation.
for every open set Ω ′ ⋐ Ω.We take the chance to provide a new proof of this bound, given that there is a minor gap in the proof of [20, Proposition 4.4, eq.(4.4)].We notice that, by [20, Proposition 4.4, eq. ( 4. 3)] and by (8), we obtain This implies that the diffuse part of the λ-pairing satisfies (A, Du) d λ = (A, Du) d , while the singular part is given by Hence, we can argue as in the proof of [20, Proposition 4.7]: we exploit [17, Theorem 3.3] to get so that, by applying ( 7) and ( 15) (restricted to Ω \ J u ), we obtain for every open set Ω ′ ⋐ Ω.
We point out that Theorem 3.5 gives a complete answer concerning the density of the pairing between A and characteristic functions of sets of finite perimeter.
A complete representation can also be given when A is a BV vector field: we state below a localization of [17, Remarks 3.4 and 3.6].
In particular, Theorem 3.5 gives a complete representation of the Cantor part of the pairing measure only if |D c u|(S A ) = 0.This requirement could be an effective restriction to the applicability of Theorem 3.5.Indeed, although L N (S A ) = 0 thanks to Lebesgue differentiation theorem, the Hausdorff dimension of S A can be equal to N (hence |D c u|(S A ) can be arbitrarily large), as it is shown in the following example.
Example 3.9.For every N ≥ 2 we construct a vector field A ∈ DM ∞ (R N ) such that div A = 0 and dim H (S A ) = N , where dim H is the Hausdorff dimension.
As a first step we exhibit a set where ∂ M E is the measure theoretic boundary of E defined in (11).
We start by considering suitable fat Cantor sets on R. Following the construction of Falconer [23,Example 4.5], for any λ ∈ (0, 1) we can construct a middle third Cantor set on [0, 1], removing at each step a proportion λ from the intervals.In this way, for any j ≥ 0 we remove from [0, 1] a family of middle open intervals {I k j } 2 j k=1 with length Let us consider the union of the intervals corresponding to even generations j: Then the fat Cantor set coincides with ∂ M E λ .Specifically, reasoning as in [17, Example 3.5], we can prove that We can now set and conclude that as claimed.Indeed, it is clear that dim H (∂ M F ) ≤ 1, and we have , there exists m large enough such that dim H (∂ M E 2 −m ) > α, for any fixed α ∈ (0, 1).This shows that dim H (∂ M F ) = 1, and so we obtain (16), by [23,Corollary 7.4].
Aiming to give a general representation of the Cantor part of the pairing measure also on S A , we are going to use the following coarea formula for the pairing, for which we refer to [17,  In the following theorem, the pairing is characterized in terms of normal traces of the field A on the level sets of u, without any assumption on S A .Theorem 3.12.
Let Z ⊂ R be the set such that for every t ∈ R \ Z the following hold: By the coarea formula in BV (Theorem 2.4), formula (13), Proposition 3.11, and Corollary 3.7, we have that L 1 (Z) = 0.
Since L 1 (Z) = 0, by [3, Proposition 3.92(a)(c)], we have that As a consequence, for |D d u|-a.e.x we have that u For every t ∈ R \ Z, let N t ⊂ ∂ * E t be a set such that the following hold: By (b) and (c), the set N t can be chosen of zero H N −1 measure.We claim that ( 19) Specifically, since the sets ∂ * E t ∩ C u , t ∈ R \ Z, are pairwise disjoint (see [25, p. 356]), we have that hence, by the coarea formula for BV functions, Finally, for every x ∈ B ∩ C u (hence, by (19), for |D d u|-a.e.x ∈ R N ), we have that x ∈ ∂ * E u(x) and ( 17) holds.
Remark 3.13.As a consequence of Theorem 3.12, the following new representation formula holds for the Cantor part of the pairing measure: In Section 4 we will provide more explicit representations of Tr * (A, ∂ * {u > u(•)})(•) (Remark 4.11 and Corollary 4.13).

Other representation formulas
For vector fields with L 1 divergence, an explicit representation of the density θ(A, Du, x) in terms of cylindrical averages has been proposed in the unpublished paper [7].
More precisely, in [7,Theorem 3.6] it is established that, if div A ∈ L 1 (Ω) and u ∈ BV (Ω) ∩ L ∞ (Ω), then whenever the limits exist, with We will extend this result by adapting the arguments of the proof contained in [7] to the general case by means of properties of the pairing recently obtained in [17].We will obtain (see Theorem 4.10 below) that the same formula holds for a general divergence-measure field A by assuming the weaker condition H N −1 (Θ A ∩ J u ) = 0, where Θ A is the jump set of div A defined in (3).
If div A has a non-vanishing jump part concentrated on the jump set J u of u, the relation (20) is no longer valid, as it is shown in the following example.
Example 4.1.Let A ∈ DM ∞ (R 2 ) be the BV vector field A(x) = a χ B 1 (x), where a ∈ R 2 is a fixed vector, and let u := χ B 1 ∈ BV (R 2 ).By Corollary 3.8 we have that On the other hand, for every x ∈ ∂B 1 and every 0 < r ≪ ρ, an explicit computation gives Cr,ρ(x,ν) However, we notice that it is still possible to achieve a representation for the pairing density θ(A, Du, x) even in the case in which H N −1 (Θ A ∩ J u ) > 0: to this purpose we will exploit the averages on half-balls introduced in [35], see Theorem 4.13 below.
For the reader's benefit, we state below the general Gauss-Green formulas for essentially bounded divergence-measure, sets of finite perimeters and functions of bounded variation, which is the localized version of [20,Theorem 6.3] in the case λ ≡ 1/2, and whose proof we leave to the interested reader.
Then the following Gauss-Green formulas hold: where E 1 is the measure theoretic interior of E.
In the particular case of u ≡ 1, Theorem 4.2 reduces to div and these formulas are fundamental tools needed in order to generalize (20) (see the proof of Lemma 4.3 below).The proof of ( 23) and ( 24) can be found for instance in [15, Theorems 4.1 and 4.2].
The following technical lemma, generalizing [7,Theorem 3.3] in the case of a singular measure div A, gives an estimate of the gap between the local behavior of the mean values of the normal traces of A on a smooth surface Σ and their analogous computed on tangent hyperplanes T x Σ to Σ.By means of the Gauss-Green formulas, we show that the gap is possibly due to the concentration of the measure | div A| on Σ. Lemma 4.3.Let A ∈ DM ∞ loc (Ω), and let Σ ⊂ R N be an oriented C 1 hypersurface with classical normal vector field ν Σ .Then, for every x ∈ Σ ∩ Ω, Tr i,e (A, Tr i,e (A, Σ)(y) dH N −1 (y) where Tr i,e (A, •) denotes either Tr i (A, •) or Tr e (A, •), and T x Σ is the tangent hyperplane to Σ in x.
Proof.Let us prove (25) for Tr i , being the computation for Tr e entirely similar.Up to a change of coordinates, we may assume that x = 0 ∈ Ω, ν Σ (0) = −e N , so that Σ is locally the graph of a C 1 function ϕ : R N −1 → R with ϕ(0) = 0, ∇ϕ(0) = 0.For every ρ > 0 such that B 2ρ ⋐ Ω, we denote Now we apply the Gauss-Green formula (23) to A and to the open piecewise Lipschitz set E ρ , and we obtain div since ) = 0 and thanks to the locality of the normal traces (6).Similarly, we apply the Gauss-Green formula (23) to A and to B − ρ , so that we get , and again thanks to (6).From ( 26) and ( 27) we obtain that where in the last inequality we have used the facts that |µ(B) − µ(C)| ≤ |µ|(B △ C) for every signed measure µ and Tr i (A, ∂B ρ ) L ∞ (∂Bρ,H N−1 ∂Bρ) ≤ A L ∞ (B 2ρ ;R N ) , thanks to (7).Since Σ is a C 1 hypersurface, then lim and, since ∇ϕ(0) = 0, we have that so that, substituting in (28), and hence (25) follows.
We briefly recall the behaviour of the cylindrical averages on hyperplanes.Tr i (A, T )(y) dH N −1 (y), which ends the proof.
Finally, we are able to specify where the cylindrical averages of the field A on oriented rectifiable sets coincide with its weak normal traces.
Proof.As a first step, let us prove the theorem in the case Σ is an oriented C 1 hypersurface, with (classical) normal vector field ν Σ .
Let x ∈ Ω ∩ Σ \ Θ A be a Lebesgue point for both Tr i (A, Σ) and Tr e (A, Σ) with respect to H N −1 Σ.From Lemma 4.3, and recalling the definition (3) of Θ A , we deduce that there exist the limits (29) lim Tr i,e (A, T x Σ)(y) dH N −1 (y) = Tr i,e (A, Σ)(x) .
On the other hand, it holds that Tr i (A, Σ) = Tr e (A, Σ) Specifically, by (8), we have that From (29) and Lemma 4.4 we deduce that, for hence the claim is proved.The general case with Σ ⊂ R N oriented countably H N −1 -rectifiable set follows directly from the previous step and the definition of weak normal trace on Σ (Section 2.3).Corollary 4.6.Let A ∈ DM ∞ loc (Ω) and let E be a set of locally finite perimeter in Ω.Then for |Dχ E |-a.e.x ∈ Ω \ Θ A the limit A(y) • ν E (x) dy exists, and Remark 4.7.This result has been proved in [7,Theorem 3.5] under the stronger assumption div A ∈ L 1 (Ω).
Proof.It is a consequence of Theorem 4.5, with Σ = ∂ * E, and of Theorem 3.12.
Theorem 4.9.Let A ∈ DM ∞ loc (Ω) and u ∈ BV loc (Ω) ∩ L 1 loc (Ω, | div A|).Let E be a set with locally finite perimeter in Ω such that supp(χ E u) ⋐ Ω. Assume that the traces u i , u e of u on Then the following Gauss-Green formulas hold: Finally, we obtain the following generalization of [7,Theorem 3.6].
If in addition we assume that (31) where A is the zero extension of A to R N \ Ω.
In particular, Proof.The first statement follows from Theorems 3.12 and 4.12.More precisely, if x ∈ J u we use (18) and Theorem 4.12 with Σ = J u , whereas if x ∈ C u we use (17) and Theorem 4.12 with Σ = ∂ * {u > u(x)}.
Remark 4.14.In light of Remark 3.14, we may exploit Theorem 4.12 as in the proof of Corollary 4.13 in order to get

Tangential properties of the pairing measure
As a consequence of the representation formula in Theorem 3.12 we easily recover the local structure of the pairing measure by means of its tangent measures.
For every x ∈ Ω, let I x,r (y) := (y − x)/r denote the homothety with scaling factor r mapping x in 0. For r > 0 small enough such that B r (x) ⋐ Ω, the pushforward I x,r # µ of a Radon measure µ is the measure acting on a test function φ ∈ C c (B 1 ) as Definition 5.1 (Tangent measures).Let µ ∈ M loc (Ω).We say that γ is a tangent measure of µ at x ∈ Ω if γ is a non-zero Radon measure and there exists some sequence (r i ) satisfying r i ↓ 0 and such that 1 |µ|(B r i (x))) I x,r i # µ * ⇀ γ in M loc (B 1 ).
We denote by Tan(µ, x) the set of all tangent measures of µ at x.For every α ≥ 0, we denote by Tan α (µ, x) the family of non-zero Radon measures γ such that there exists a sequence r i ↓ 0 for which Following the notation established in Section 2, in the following results for any given function f ∈ L 1 loc (Ω, |µ|) we use the notation f (x) := f (x) for every x ∈ Ω Lebesgue point of f with respect to |µ|.We start by proving the following property of the tangent measures (for related results see [3,Theorem 2.44] and [30,Lemma 14.6]).Proof.Let x ∈ Ω be as in the statement, and let r i ↓ 0 (so that B r i (x) ⋐ Ω) be such that at least one of the sequences c i I x,r i # (f µ) and c i I x,r i # µ converges weakly * to a Radon measure, where c i = 1 |µ|(Br i (x))) .To fix the ideas, assume that c i I x,r i # µ * ⇀ γ.For every ϕ ∈ C c (B 1 ), we have that and the right-hand side converges to 0 as i → +∞ since x is a Lebesgue point of f with respect to µ.It follows that the sequence c i I x,r i # (f µ) converges weakly * to f (x) γ, so that we can conclude that Tan(f µ, x) = f (x) Tan(µ, x).
The equality for Tan α can be proved by dealing separately with the case in which both tangent sets are empty and the one in which they are not.In the latter we have |µ|(B r (x)) ≤ Cr α for some C ≥ 0, so that one can argue as above.
We can now state two results on the tangent measures of pairings.To this purpose, for any unit vector ν we set ν ⊥ := {y ∈ R N : y • ν = 0}.If instead x ∈ J u is a Lebesgue point of Tr * (A, J u ) with respect to |D j u| such that Tr * (A, J u )(x) = 0, then Tan N −1 ((A, Du) j , x) = Tr * (A, J u )(x) [u + (x) − u − (x)] H N −1 ν ⊥ u (x).Proof.It is a consequence of Lemma 5.2, Theorem 3.12 and the Federer-Vol'pert theorem (see [3,Theorem 3.78]).
θ(A, Du, x) = A(x) • ∇u(x) |∇u(x)| =: Tr(A, {u = u(x)}) , if ∇u(x) = 0, where Tr(A, {u = u(x)}) denotes the normal trace of A on the regular level set {u = u(x)}.In the general case, the usual decomposition Du = ∇u L N + D j u + D c u of the measure Du into its absolutely continuous, jump and Cantor parts, leads to a corresponding decomposition of the pairing measure (A, Du) = (A, Du) a + (A, Du) j + (A, Du) c .