Pairings between bounded divergence-measure vector fields and BV functions

TL;DR

This paper introduces a new family of pairings between divergence-measure vector fields and BV functions, extending existing concepts and analyzing their properties, including semicontinuity under strict convergence.

Contribution

It develops a generalized framework for pairings that depend on the pointwise representative of BV functions, preserving key properties and characterizing semicontinuous cases.

Findings

New family of pairings introduced

Standard pairing does not always have semicontinuity

Characterization of semicontinuous pairings

Abstract

We introduce a family of pairings between a bounded divergence-measure vector field and a function $u$ of bounded variation, depending on the choice of the pointwise representative of $u$. We prove that these pairings inherit from the standard one, introduced in [6,10], all the main properties and features (e.g. coarea, Leibniz and Gauss--Green formulas). We also characterize the pairings making the corresponding functionals semicontinuous with respect to the strict convergence in $BV$. We remark that the standard pairing in general does not share this property.

Equations

$$\mathrm{div}(u\mathbf{A})=u\mathrm{div}\mathbf{A}+\mathbf{A}\cdot \nabla u,$$

$$\mathrm{div}(u\mathbf{A})=u^{\ast }\mathrm{div}\mathbf{A}+{\left(\mathbf{A},Du\right)}_{\ast }$$

$${\left(\mathbf{A},Du\right)}_1:=-u^{+}\mathrm{div}\mathbf{A}+\mathrm{div}(u\mathbf{A})$$

$$\mathrm{div}(u\mathbf{A})=u^{\lambda }\mathrm{div}\mathbf{A}+{\left(\mathbf{A},Du\right)}_{\lambda },$$

$$F_{\phi }(u):=⟨{\left(\mathbf{A},Du\right)}_{\lambda },\phi ⟩,\phi \in C_c({\mathbb{R}}^N),\phi \ge 0.$$

$${\left(\mathbf{A},Du\right)}_{\lambda }=-u^{+}(\mathrm{div}\mathbf{A}{)}^{+}+u^{-}(\mathrm{div}\mathbf{A}{)}^{-}+\mathrm{div}(u\mathbf{A}),$$

$${\left(\mathbf{A},Du\right)}_{\lambda }=-u^{-}(\mathrm{div}\mathbf{A}{)}^{+}+u^{+}(\mathrm{div}\mathbf{A}{)}^{-}+\mathrm{div}(u\mathbf{A}).$$

$$E^t:=\left\{x\in {\mathbb{R}}^N:\underset{\rho \to 0^{+}}{\lim }\frac{{\mathcal{L}}^N(E\cap B_{\rho }(x))}{{\mathcal{L}}^N(B_{\rho }(x))}=t\right\}$$

$$u^{+}(x):=\inf \{t\in \mathbb{R}:\{u>t\}\text{has density 0 at x}\},$$

$$u^{-}(x):=\sup \{t\in \mathbb{R}:\{u>t\}\text{has density 1 at x}\}.$$

$$\underset{r\to 0^{+}}{\lim }\frac{1}{{\mathcal{L}}^N\left(B_r(x)\right)}{\int }_{B_r\left(x\right)}\left|u(y)-z\right|dy=0.$$

$$\begin{array}{c}\underset{r\to 0^{+}}{\lim }\frac{1}{{\mathcal{L}}^N(B_r^i(x))}{\int }_{B_r^i(x)}|u(y)-a|dy=0,\\ \underset{r\to 0^{+}}{\lim }\frac{1}{{\mathcal{L}}^N(B_r^e(x))}{\int }_{B_r^e(x)}|u(y)-b|dy=0,\end{array}$$

$$|\mu |(E):=\sup \left\{\sum _{h=0}^{\infty }|\mu (E_h)|:E_h\text{μ-measurable sets, pairwise disjoint},E=\bigcup _{h=0}^{\infty }{E}_h\right\},$$

$${\mu }^{+}:=\frac{|\mu |+\mu }{2},{\mu }^{-}:=\frac{|\mu |-\mu }{2}.$$

$$\mu \text{vrule}height=7.0pt,width=0.5pt,depth=0.0pt\text{vrule}height=0.5pt,width=6.0pt,depth=0.0ptE(B)=\mu (E\cap B),\forall B\text{μ-measurable},B\subset \Omega .$$

$$E\subset \Omega ,|\mu |(E)=0⟹|\mu |(B_r(x))=o(r^{N-1})\text{for HN−1–a.e.}x\in E.$$

$$\exists \underset{r\searrow 0}{\lim }\frac{|\mu |(B_r(x))}{r^{N-1}}=L\in \mathbb{R}⟺\exists \underset{r\searrow 0}{\lim }\frac{\mu (B_r(x))}{r^{N-1}}=\psi (x)L.$$

$$|\psi (x)|=1,\underset{r\searrow 0}{\lim }\frac{1}{|\mu |(B_r(x))}{\int }_{B_r(x)}|\psi (y)-\psi (x)|d|\mu |=0,\forall x\in A.$$

$$Z_1:=\left\{x\in \Omega :\underset{r\searrow 0}{\mathrm{limsup}}\frac{|\mu |(B_r(x))}{r^{N-1}}=+\infty \right\}$$

$$\begin{array}{rl}\left|{\int }_{\Omega }\phi d|\mu |-\psi (x){\int }_{\Omega }\phi d\mu \right|& =\left|{\int }_{B_r(x)}\phi (y)[1-\psi (y)\psi (x)]d|\mu |(y)\right|\\ & \le \Vert \phi {\Vert }_{\infty }|\mu |(B_r(x))-{\int }_{B_r(x)}|\psi (y)-\psi (x)|d|\mu |(y).\end{array}$$

$$\left|\frac{|\mu |(B_r(x))}{r^{N-1}}-\psi (x)\frac{\mu (B_r(x))}{r^{N-1}}\right|\le \frac{|\mu |(B_r(x))}{r^{N-1}}-{\int }_{B_r(x)}|\psi (y)-\psi (x)|d|\mu |(y),$$

$${\Theta }_{\mu }:=\left\{x\in \Omega :\underset{r\to 0+}{\mathrm{limsup}}\frac{|\mu |(B_r(x))}{r^{N-1}}>0\right\}$$

$$Du=D^{a}u+D^{s}u,D^{a}u=\nabla u{\mathcal{L}}^N,$$

$$D^{j}u:=D^{s}u\text{vrule}height=7.0pt,width=0.5pt,depth=0.0pt\text{vrule}height=0.5pt,width=6.0pt,depth=0.0pt{J}_u=(u^i-u^e){\nu }_u{\mathcal{H}}^{N-1}\text{vrule}height=7.0pt,width=0.5pt,depth=0.0pt\text{vrule}height=0.5pt,width=6.0pt,depth=0.0pt{J}_u.$$

$$u^{-}(x)=\min \{u^i(x),u^e(x)\},u^{+}(x)=\max \{u^i(x),u^e(x)\},x\in J_u.$$

$$u^i=u^e=\tilde{u}\text{in}\Omega \setminus S_u.$$

$$u^{\lambda }(x):=\{\begin{array}{ll}\tilde{u}(x),& x\in \Omega \setminus S_u,\\ (1-\lambda (x))u^{-}(x)+\lambda (x)u^{+}(x),& x\in J_u.\end{array}$$

$$P(E,\Omega ):=\sup \left\{{\int }_E\mathrm{div}\phi dx:\phi \in C_c^1(\Omega ,{\mathbb{R}}^N),\Vert \phi {\Vert }_{\infty }\le 1\right\}.$$

$${\tilde{\nu }}_E(x):=\underset{\rho \to 0^{+}}{\lim }\frac{D{\chi }_E(B_{\rho }(x))}{|D{\chi }_E|(B_{\rho }(x))}$$

$${\int }_{\Omega }\phi d\mathrm{div}\mathbf{A}=-{\int }_{\Omega }\mathbf{A}\cdot \nabla \phi dx\forall \phi \in C_c^{\infty }(\Omega ).$$

$$⟨\mathrm{Tr}(\mathbf{A},\partial {\Omega }^{\prime }),\phi ⟩:={\int }_{{\Omega }^{\prime }}\mathbf{A}\cdot \nabla \phi dx+{\int }_{{\Omega }^{\prime }}\phi d\mathrm{div}\mathbf{A},\forall \phi \in C_c^{\infty }(\Omega ).$$

Full text

1. 1 Introduction

2. 2 Notation and preliminary results

3. 2.1 Measures

4. 2.2 Functions of bounded variation

5. 2.3 Divergence–measure fields

6. 2.4 Weak normal traces

7. 3 Some remarks on $L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$

8. 4 Definition and basic properties of pairings

9. 5 Coarea formula for generalized pairings

10. 6 Chain rule, Leibniz and Gauss–Green formulas for generalized pairings

11. 7 Semicontinuity results

1. Introduction

In the seminal papers [Anz, ChenFrid], the product rule

[TABLE]

for smooth functions $u$ and regular vector fields $\boldsymbol{A}$ in $\mathbb{R}^{N}$, has been suitably extended to $BV$ functions and bounded divergence-measure vector fields. In particular, Chen and Frid [ChenFrid] showed, using a regularization argument, that there exists a finite Radon measure $\left(\boldsymbol{A},Du\right)_{*}$, which coincides to $\boldsymbol{A}\cdot\nabla u\,{\mathcal{L}}^{N}$ in the smooth case, such that the relation

[TABLE]

holds in the sense of measures. The measure $\left(\boldsymbol{A},Du\right)_{*}$, usually called Anzellotti’s pairing and that we call in the sequel the standard pairing between $\boldsymbol{A}$ and $Du$, is then defined in terms of the precise representative $u^{*}$ of $u$, which is the pointwise value of $u$ obtained as limit of regularizations by convolutions.

The standard pairing turns out to be a basic tool in many applications. We mention here, among others: extensions of the Gauss–Green formula [Anz, Cas, ChCoTo, ChTo, ChToZi, ComiMag, ComiPayne, CD3, CD4, LeoSar]; the setting of the Euler–Lagrange equations associated with integral functionals defined in $BV$ [ABCM, MaRoSe, Mazon2016]; Dirichlet problems for equations involving the $1$–Laplace operator [K1, HI, AVCM, Cas, DeGiSe, DeGiOlPe]; conservation laws [ChFr1, ChenFrid, ChTo2, ChTo, ChToZi, CD2]; the Prescribed Mean Curvature problem and capillarity [LeoSar, LeoSar2]; continuum mechanics [ChCoTo, DGMM, Silh, Schu].

On the other hand, the standard pairing is not adequate when dealing with obstacle problems in $BV$ (see [SchSch, SchSch2, SchSch3]) or with semicontinuity properties, as we will explain below. The aim of this paper is to introduce a new family of pairings, depending on the choice of the pointwise representative of $u$, suitable to treat this kind of problems.

The main ingredients to build this family of pairings are the absolute continuity of the measure $\operatorname{div}\boldsymbol{A}$ with respect to the $(N-1)$-dimensional Hausdorff measure ${\mathcal{H}}^{N-1}$, and the fact that the pointwise value of a $BV$ function can be specified up to a ${\mathcal{H}}^{N-1}$-negligible set. Indeed, a $BV$ function $u$ is approximately continuous outside a singular set $S_{u}$ and its approximate upper and lower limits $u^{+}$ and $u^{-}$ coincide with the traces of $u$ on the countably ${\mathcal{H}}^{N-1}$-rectifiable jump set $J_{u}\subset S_{u}$, with ${\mathcal{H}}^{N-1}(S_{u}\setminus J_{u})=0$ (see Section 2.2). Hence, a representative of $u$ can be defined by its approximate limit $\tilde{u}$ outside $S_{u}$ and through its traces $u^{\pm}$ on $J_{u}$. We remark again that the presence of $u^{*}:=(u^{+}+u^{-})/2$ in (2) as the pointwise representative of $u$ is due to the regularization argument used in [ChenFrid] in order to define the standard pairing.

Recently, Scheven and Schmidt [SchSch, SchSch2, SchSch3] have been in need to introduce the pairing

[TABLE]

in order to study weakly $1$-superharmonic functions and minimization problems for the total variation with an obstacle. Indeed, in this case, the presence of the representative $u^{+}$ comes out from (1) using the one-sided approximation procedure of $u$ introduced in [CDLP].

In this paper we prove that, for every Borel function $\lambda\colon\mathbb{R}^{N}\to[0,1]$, there exists a measure $\left(\boldsymbol{A},Du\right)_{\lambda}$ such that

[TABLE]

where $u^{\lambda}:=(1-\lambda)u^{-}+\lambda u^{+}$ is a selection of the multifunction $x\mapsto[u^{-}(x),u^{+}(x)]$. We show that, if the jump part $\operatorname{div}^{j}\boldsymbol{A}$ of $\operatorname{div}\boldsymbol{A}$ vanishes (see Proposition 2.3 for the definition), then $\left(\boldsymbol{A},Du\right)_{\lambda}$ is independent of $\lambda$.

We show that this freedom in the choice of $u^{\lambda}$ is necessary in order to obtain semicontinuity results in $BV$ for the functionals

[TABLE]

We characterize the selections $\lambda$ such that these functionals are lower (resp. upper) semicontinuous with respect to the strict convergence in $BV$. More precisely, denoting by $(\operatorname{div}\boldsymbol{A})^{\pm}$ the positive and the negative part of the measure $\operatorname{div}\boldsymbol{A}$, the choices of $\lambda$ which guarantee the lower semicontinuity of the functionals in (5) satisfy

[TABLE]

whereas the upper semicontinuity is characterized by

[TABLE]

As a consequence, it is a matter of fact that, in general, the standard pairing does not share these semicontinuity properties. On the other hand, if $\operatorname{div}\boldsymbol{A}\leq 0$, as in [SchSch, SchSch2, SchSch3], from the above result follows that the pairing (3) is upper semicontinuous with respect to the strict convergence in $BV$.

The plan of the paper is the following. In Section 2 we recall some known results on $BV$ functions, divergence-measure vector fields and their weak normal traces. In Section 3 we focus our attention on the summability of $u^{\lambda}$ with respect to the measure $|\operatorname{div}\boldsymbol{A}|$ and on some related properties of the truncated functions. In Sections 4, 5 and 6 we introduce the generalized pairing and we prove that it inherits from the standard one all the main properties and features. More precisely, $\left(\boldsymbol{A},Du\right)_{\lambda}$ is a Radon measure, absolutely continuous with respect to $|Du|$, it satisfies the coarea, the chain rule and the Leibniz formulas, and it is consistent with the Gauss–Green formula.

The proofs of these results are based on the analogous properties valid for the standard pairing (see [CD3]), the fact that the generalized pairing differs from the standard one only by a term concentrated on $J_{u}$ (see (20)), and some representation results of the normal traces of $\boldsymbol{A}$ on $J_{u}$ (see [AmbCriMan]).

Our main application of the above theory is proposed in Section 7, where we consider the semicontinuity properties of the functionals $F_{\varphi}$ defined in (5), with respect to the strict convergence in $BV$. In Theorem 7.6 we prove the characterizations (6)–(7) of the semicontinuous pairings. The proof is based on a recent result of Lahti (see [La]), which assures the lower (upper) semicontinuity of the lower $u^{-}$ (upper $u^{+}$) limit under the strict convergence in $BV$, combined with the one-sided approximation result in [CDLP], and a very careful treatment of the jump part of the measure $\operatorname{div}\boldsymbol{A}$. We show by easy examples that no semicontinuity property has to be expected with respect to the weak∗ convergence in $BV$.

2. Notation and preliminary results

In the following $\Omega$ will always denote a nonempty open subset of $\mathbb{R}^{N}$. For every $E\subset\Omega$, $\chi_{E}$ denotes its characteristic function. We say that $E_{h}$ converges to $E$ if $\chi_{E_{h}}$ converges to $\chi_{E}$ in $L^{1}(\Omega)$.

We denote by ${\mathcal{L}}^{N}$ and ${\mathcal{H}}^{N-1}$ the Lebesgue measure and the $(N-1)$–dimensional Hausdorff measure in $\mathbb{R}^{N}$, respectively.

If $E\subset\mathbb{R}^{N}$ is an open set, the notation $\varphi\nearrow\chi_{E}$ denotes any family $(\varphi_{j})$ of smooth functions with support in $E$, such that $0\leq\varphi_{j}\leq 1$, and $\lim_{j}\varphi_{j}(x)=1$ for every $x\in E$.

Given an ${\mathcal{L}}^{N}$-measurable set $E\subset\mathbb{R}^{N}$, For every $t\in[0,1]$ we denote by $E^{t}$ the set

[TABLE]

of all points where $E$ has density $t$. The sets $E^{0}$, $E^{1}$, $\partial^{e}E:=\mathbb{R}^{N}\setminus(E^{0}\cup E^{1})$ are called respectively the measure theoretic exterior, the measure theoretic interior and the essential boundary of $E$.

Let $u\colon\Omega\to\mathbb{R}$ be a Borel function. We denote by $u^{-}$ and $u^{+}$ the approximate lower limit and the approximate upper limit of $u$, defined respectively by

[TABLE]

The function $u$ is approximately continuous at $x\in\Omega$ if $u^{+}(x)=u^{-}(x)$ and, in this case, we denote by $\widetilde{u}(x)$ the common value.

Given $u\in L^{1}_{{\rm loc}}(\Omega)$, $x\in\Omega$ is a Lebesgue point of $u$ (with respect to ${\mathcal{L}}^{N}$) if there exists $z\in\mathbb{R}$ such that

[TABLE]

In this case, $x$ is a point of approximate continuity, and $z=\widetilde{u}(x)$ (see [FonLeoBook, Proposition 1.163]). We denote by $S_{u}\subset\Omega$ the set of points where this property does not hold.

We say that $x\in\Omega$ is an approximate jump point of $u$ if there exist $a,b\in\mathbb{R}$ and a unit vector $\nu\in\mathbb{R}^{n}$ such that $a\neq b$ and

[TABLE]

where $B_{r}^{i}(x):=\{y\in B_{r}(x):\ (y-x)\cdot\nu>0\}$, and $B_{r}^{e}(x):=\{y\in B_{r}(x):\ (y-x)\cdot\nu<0\}$. The triplet $(a,b,\nu)$, uniquely determined by (8) up to a permutation of $(a,b)$ and a simultaneous change of sign of $\nu$, is denoted by $({u^{i}}(x),{u^{e}}(x),\nu_{u}(x))$. The set of approximate jump points of $u$ will be denoted by $J_{u}$.

A ${\mathcal{H}}^{N-1}$-measurable set $E\subset\mathbb{R}^{N}$ is countably ${\mathcal{H}}^{N-1}$-rectifiable if there exist countably many $C^{1}$ graphs $(\Sigma_{i})_{i\in\mathbb{N}}$ such that ${\mathcal{H}}^{N-1}\left(E\setminus\bigcup_{i}\Sigma_{i}\right)=0$.

2.1. Measures

The space of all Radon measures on $\Omega$ will be denoted by $\mathcal{M}(\Omega)$.

Given $\mu\in\mathcal{M}(\Omega)$, its total variation $|\mu|$ is the nonnegative Radon measure defined by

[TABLE]

for every $\mu$-measurable set $E$ and its positive and negative parts are defined, respectively, by

[TABLE]

If $\mu_{1},\mu_{2}\in\mathcal{M}(\Omega)$, then $\max\{\mu_{1},\mu_{2}\}$ (resp. $\min\{\mu_{1},\mu_{2}\}$) is the measure that assigns to every Borel set $E\subset\Omega$, the supremum (resp. infimum) of $\mu_{1}(E_{1})+\mu_{2}(E_{2})$ among all pairwise disjoint Borel sets $E_{1},E_{2}$ such that $E_{1}\cup E_{2}=E$.

Given $\mu\in\mathcal{M}(\Omega)$ and a $\mu$-measurable set $E$, the restriction $\mu\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits E$ is the Radon measure defined by

[TABLE]

We recall the following property (see [AFP], Proposition 2.56 and formula (2.41)):

[TABLE]

Given a nonnegative Borel measure $\nu$, we say that $\mu\in\mathcal{M}(\Omega)$ is absolutely continuous with respect to $\nu$ (and we write $\mu\ll\nu$), if $|\mu|(B)=0$ for every set $B$ such that $\nu(B)=0$.

We say that two positive measures $\nu_{1}$, $\nu_{2}\in\mathcal{M}(\Omega)$ are mutually singular (and we write $\nu_{1}\perp\nu_{2}$) if there exists a Borel set $E$ such that $|\nu_{1}|(E)=0$ and $|\nu_{2}|(\Omega\setminus E)=0$.

By the Radon–Nikodým theorem, given a nonnegative Radon measure $\nu$, every $\mu\in\mathcal{M}(\Omega)$ can be uniquely decomposed as $\mu=\mu_{1}+\mu_{2}$ with $\mu_{1}\ll\nu$ and $\mu_{2}\perp\nu$, and there exists a unique function (called the density of $\mu$ with respect to $\nu$) $\psi_{\nu}\in L^{1}(\Omega,\nu)$ such that $\mu_{1}=\psi_{\nu}\nu$. In particular, since $\mu\ll|\mu|$, then there exists $\psi\in L^{1}(\Omega,|\mu|)$, with $|\psi|=1$ $|\mu|$–a.e. in $\Omega$, and such that $\mu=\psi|\mu|$. This is usually called the polar decomposition of $\mu$.

The following lemma shows the relation between the densities of $\mu$ and $|\mu|$, where $\mu$ is a Radon measure absolutely continuous with respect to ${\mathcal{H}}^{N-1}$.

Lemma 2.1**.**

Let $\mu\ll{\mathcal{H}}^{N-1}$ be a Radon measure in $\Omega$, and let $\mu=\psi|\mu|$ be its polar decomposition. Then there exists a Borel set $Z\subset\Omega$, with $|\mu|(Z)=0$, such that every $x\in\Omega\setminus Z$ is a Lebesgue point of $\psi$ with respect to $|\mu|$, and

[TABLE]

Proof.

Let $A\subset\Omega$ be the set of Lebesgue points of $\psi$ with respect to $|\mu|$. By [AFP, Corollary 2.23], we have that $|\mu|(\Omega\setminus A)=0$. Since $|\psi|=1$ $|\mu|$-a.e., it is not restrictive to assume that

[TABLE]

Moreover, from [AFP, Theorem 2.56 and (2.40)], the set

[TABLE]

has zero ${\mathcal{H}}^{N-1}$-measure, hence also $|\mu|(Z_{1})=0$.

If we set $Z:=(\Omega\setminus A)\cup Z_{1}$, then $|\mu|(Z)=0$ and (10) holds in $\Omega\setminus Z$. Specifically, given $x\in\Omega\setminus Z$, $B_{r}(x)\subset\Omega$ and $\varphi\in C_{c}(\Omega)$ with support in $B_{r}(x)$, since $|\psi(x)|=1$, we have that $|1-\psi(y)\psi(x)|=|\psi(y)-\psi(x)|$, and hence

[TABLE]

Taking $\varphi\nearrow\chi_{B_{r}(x)}$ and dividing by $r^{N-1}$ we finally get

[TABLE]

hence (10) follows because $x\not\in Z_{1}$ and $x$ is a Lebesgue point of $\psi$. ∎

Given $\mu\in\mathcal{M}(\Omega)$, we denote by $\mu=\mu^{a}+\mu^{s}$ its Lebesgue decomposition in the absolutely continuous part $\mu^{a}\ll{\mathcal{L}}^{N}$ and the singular part $\mu^{s}\perp{\mathcal{L}}^{N}$. We recall a relevant decomposition result for $\mu^{s}$ (see [ADM], Proposition 5).

Proposition 2.2**.**

If $\mu\in\mathcal{M}(\Omega)$ is such that $\mu^{s}\ll{\mathcal{H}}^{N-1}$, then $\mu^{s}$ can be uniquely decomposed as the sum $\mu^{j}+\mu^{c}$, where $\mu^{j}$, $\mu^{c}\in\mathcal{M}(\Omega)$ are two mutually singular measures having the following properties:

• (i)

$\mu^{c}(B)=0$* for every $B$ such that ${\mathcal{H}}^{N-1}(B)<+\infty$;*

• (ii)

the set

[TABLE]

is a Borel set, $\sigma$–finite with respect to ${\mathcal{H}}^{N-1}$;

• (iii)

there exists $f\in L^{1}(\Theta_{\mu},{\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Theta_{\mu})$ such that $\mu^{j}=f\,{\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Theta_{\mu}$.

The measures $\mu^{j}$, $\mu^{c}$ are called jump part and Cantor part of the measure $\mu$, while $\Theta_{\mu}$ is called jump set of $\mu$.

2.2. Functions of bounded variation

We say that $u\in L^{1}(\Omega)$ is a function of bounded variation in $\Omega$ if the distributional derivative $Du$ of $u$ is a finite Radon measure in $\Omega$. The vector space of all functions of bounded variation in $\Omega$ will be denoted by $BV(\Omega)$. Moreover, we will denote by $BV_{{\rm loc}}(\Omega)$ the set of functions $u\in L^{1}_{{\rm loc}}(\Omega)$ that belongs to $BV(A)$ for every open set $A\Subset\Omega$ (i.e., the closure $\overline{A}$ of $A$ is a compact subset of $\Omega$).

If $u\in BV(\Omega)$, then $Du$ can be decomposed as the sum of the absolutely continuous and the singular part with respect to the Lebesgue measure, i.e.

[TABLE]

where $\nabla u$ is the approximate gradient of $u$, defined ${\mathcal{L}}^{N}$-a.e. in $\Omega$ (see [AFP, Section 3.9]). The jump set $J_{u}$ has the following properties: it is countably $\mathcal{H}^{N-1}$–rectifiable and $\mathcal{H}^{N-1}(S_{u}\setminus J_{u})=0$ (see [AFP, Definition 2.57 and Theorem 3.78]); it is contained in the set $\Theta_{Du}$ defined in Proposition 2.2(ii) with $\mu=Du$, and ${\mathcal{H}}^{N-1}(\Theta_{Du}\setminus J_{u})=0$ (see [AFP, Proposition 3.92(b)]). By Proposition 2.2, the singular part $D^{s}u$ can be further decomposed as the sum of its Cantor and jump part, i.e. $D^{s}u=D^{c}u+D^{j}u$, $D^{c}u:=D^{s}u\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits(\Omega\setminus S_{u})$, and

[TABLE]

We denote by $D^{d}u:=D^{a}u+D^{c}u$ the diffuse part of the measure $Du$.

At every point $x\in J_{u}$ we have that $-\infty<u^{-}(x)<u^{+}(x)<+\infty$ and

[TABLE]

Moreover, we can always choose an orientation on $J_{u}$ such that ${u^{i}}=u^{+}$ on $J_{u}$ (see [GMS1, §4.1.4, Theorem 2]). In the following we shall always extend the functions ${u^{i}},{u^{e}}$ to $\Omega\setminus(S_{u}\setminus J_{u})$ by setting

[TABLE]

Given a Borel function $\lambda\colon\Omega\to[0,1]$, the $\lambda$–representative of $u\in BV_{\rm loc}(\Omega)$ is defined by

[TABLE]

When $\lambda(x)=1/2$ for every $x\in\Omega$, the $\lambda$–representative coincides with the precise representative $u^{*}:=(u^{+}+u^{-})/2$ of $u$.

Let $E$ be an ${\mathcal{L}}^{N}$-measurable subset of $\mathbb{R}^{N}$. For every open set $\Omega\subset\mathbb{R}^{N}$ the perimeter $P(E,\Omega)$ is defined by

[TABLE]

We say that $E$ is of finite perimeter in $\Omega$ if $P(E,\Omega)<+\infty$.

Denoting by $\chi_{E}$ the characteristic function of $E$, if $E$ is a set of finite perimeter in $\Omega$, then $D\chi_{E}$ is a finite Radon measure in $\Omega$ and $P(E,\Omega)=|D\chi_{E}|(\Omega)$.

If $\Omega\subset\mathbb{R}^{N}$ is the largest open set such that $E$ is locally of finite perimeter in $\Omega$, we call reduced boundary $\partial^{*}E$ of $E$ the set of all points $x\in\Omega$ in the support of $|D\chi_{E}|$ such that the limit

[TABLE]

exists in $\mathbb{R}^{N}$ and satisfies $|\widetilde{\nu}_{E}(x)|=1$. The function $\widetilde{\nu}_{E}\colon\partial^{*}E\to S^{N-1}$ is called the measure theoretic unit interior normal to $E$.

A fundamental result of De Giorgi (see [AFP, Theorem 3.59]) states that $\partial^{*}E$ is countably $(N-1)$-rectifiable and $|D\chi_{E}|={\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\partial^{*}E$. If $E$ has finite perimeter in $\Omega$, Federer’s structure theorem states that $\partial^{*}E\cap\Omega\subset E^{1/2}\subset\partial^{e}E$ and $\mathcal{H}^{N-1}(\Omega\setminus(E^{0}\cup\partial^{e}E\cup E^{1}))=0$ (see [AFP, Theorem 3.61]).

2.3. Divergence–measure fields

We will denote by ${\mathcal{DM}^{\infty}}(\Omega)$ the space of all vector fields $\boldsymbol{A}\in L^{\infty}(\Omega,\mathbb{R}^{N})$ whose divergence in the sense of distributions is a finite Radon measure in $\Omega$, acting as

[TABLE]

Similarly, $\mathcal{DM}^{\infty}_{{\rm loc}}{(\Omega)}$ will denote the space of all vector fields $\boldsymbol{A}\in L^{\infty}_{{\rm loc}}(\Omega,\mathbb{R}^{N})$ whose divergence in the sense of distributions is a Radon measure in $\Omega$.

The basic properties of these vector fields are collected in the following proposition.

Proposition 2.3**.**

Let $\boldsymbol{A}$ be a vector field belonging to $\mathcal{DM}^{\infty}{(\Omega)}$, and let $\Theta_{\boldsymbol{A}}$ be the jump set of the measure $\mu=|\operatorname{div}\boldsymbol{A}|$, defined in Proposition 2.2(ii). Then the following hold.

• (i)

$|\operatorname{div}\boldsymbol{A}|\ll{\mathcal{H}}^{N-1}$;

• (ii)

$\Theta_{\boldsymbol{A}}$* is a Borel set, $\sigma$-finite with respect to ${\mathcal{H}}^{N-1}$;*

• (iii)

$\operatorname{div}\boldsymbol{A}=\operatorname{div}^{a}\boldsymbol{A}+\operatorname{div}^{c}\boldsymbol{A}+\operatorname{div}^{j}\boldsymbol{A},$* where $\operatorname{div}^{a}\boldsymbol{A}$ is absolutely continuous with respect to ${\mathcal{L}}^{N}$, $\operatorname{div}^{c}\boldsymbol{A}(B)=0$ for every set $B$ with ${\mathcal{H}}^{N-1}(B)<+\infty$, and there exists $f\in L^{1}(\Theta_{\boldsymbol{A}},{\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Theta_{\boldsymbol{A}})$ such that $\operatorname{div}^{j}\boldsymbol{A}=f\,{\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Theta_{\boldsymbol{A}}$.*

Proof.

The main property (i) is proved in [ChenFrid, Proposition 3.1]. The decomposition then follows from Proposition 2.2. ∎

2.4. Weak normal traces

In what follows, we will deal with the traces of the normal component of a vector field $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ on a countably $\mathcal{H}^{N-1}$–rectifiable set $\Sigma\subset\Omega$. In order to fix the notation, we briefly recall the construction given in [AmbCriMan] (see Propositions 3.2, 3.4 and Definition 3.3).

Given a domain $\Omega^{\prime}\Subset\Omega$ of class $C^{1}$, the trace of the normal component of $\boldsymbol{A}$ on $\partial\Omega^{\prime}$ is the distribution defined by

[TABLE]

It turns out that this distribution is induced by an $L^{\infty}$ function on $\partial\Omega^{\prime}$, still denoted by $\operatorname{Tr}(\boldsymbol{A},\partial\Omega^{\prime})$, and

[TABLE]

Given a countably ${\mathcal{H}}^{N-1}$–rectifiable set $\Sigma$, there exist a covering $(\Sigma_{i})_{i\in\mathbb{N}}$ of $\Sigma$ and Borel sets $N_{i}\subseteq\Sigma_{i}$ with the following properties:

• (R1)

$\Sigma_{i}$ is an oriented $C^{1}$ hypersurface, with (classical) normal vector field $\nu_{\Sigma_{i}}$;

• (R2)

$N_{i}\subseteq\Sigma_{i}$ are pairwise disjoint Borel sets such that ${\mathcal{H}}^{N-1}(\Sigma\setminus\bigcup_{i}N_{i})=0$;

• (R3)

for every $i\in\mathbb{N}$, there exist two open bounded sets $\Omega_{i},\Omega^{\prime}_{i}$ with $C^{1}$ boundary and exterior normal vectors $\nu_{\Omega_{i}}$ and $\nu_{\Omega_{i}^{\prime}}$ respectively, such that $N_{i}\subseteq\partial\Omega_{i}\cap\partial\Omega^{\prime}_{i}$, and

[TABLE]

We can fix an orientation on $\Sigma$, given by

[TABLE]

and the normal traces of $\boldsymbol{A}$ on $\Sigma$ are defined by

[TABLE]

By a deep localization property proved in [AmbCriMan, Proposition 3.2], these definitions are independent of the choice of $\Sigma_{i}$ and $N_{i}$. In what follows, the pair $(\Sigma,\nu_{\Sigma})$ (or, simply, $\Sigma)$ will be called and oriented countably ${\mathcal{H}}^{N-1}$-rectifiable set.

We remark that, the normal traces belong to $L^{\infty}(\Sigma,\mathcal{H}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Sigma)$ and

[TABLE]

(see [AmbCriMan, Proposition 3.4]). In particular, by (13), $|\operatorname{div}\boldsymbol{A}|(\Sigma)\leq\|\boldsymbol{A}\|_{\infty}{\mathcal{H}}^{N-1}(\Sigma)$.

Remark 2.4*.*

We observe that, if $\Sigma$ is oriented by a normal vector field $\nu$ and $\Sigma^{\prime}$ is the same set oriented by $\nu^{\prime}:=-\nu$, then

[TABLE]

so that the difference $\operatorname{Tr}^{i}(\boldsymbol{A},\Sigma)-\operatorname{Tr}^{e}(\boldsymbol{A},\Sigma)$ is independent of the choice of the orientation on $\Sigma$.

The following result is a consequence of (14) and will be used in the study of the semicontinuity of the generalized pairing (see Theorem 7.6).

Theorem 2.5**.**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$, let $\operatorname{div}\boldsymbol{A}=\psi_{\boldsymbol{A}}|\operatorname{div}\boldsymbol{A}|$ be the polar decomposition of the measure $\operatorname{div}\boldsymbol{A}$, and let $\Sigma\subset\Omega$ be an oriented countably ${\mathcal{H}}^{N-1}$-rectifiable set. Then

[TABLE]

Proof.

From (9) with $\mu:=|\operatorname{div}\boldsymbol{A}|\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Sigma$ and $E:=\Omega\setminus\Sigma$, we have that

[TABLE]

On the other hand, by (14)

[TABLE]

and hence (15) holds.

Let us define the sets

[TABLE]

By Proposition 2.3(i) and [AFP, Theorems 2.22 and 2.83], we infer that ${\mathcal{H}}^{N-1}(\Sigma\setminus(\Sigma^{\prime}\cup\Sigma^{\prime\prime}))=0$. From (15) and Lemma 2.1 we deduce that the equality in (16) holds for ${\mathcal{H}}^{N-1}$-a.e.  $x\in\Sigma^{\prime\prime}$. On the other hand, from (15) we deduce that $\operatorname{Tr}^{i}(\boldsymbol{A},\Sigma)(x)-\operatorname{Tr}^{e}(\boldsymbol{A},\Sigma)(x)=0$ for ${\mathcal{H}}^{N-1}$-a.e. $x\in\Sigma^{\prime}$, hence (16) follows. ∎

For later use, we recall here a result proved in [CD3, Proposition 3.1].

Proposition 2.6**.**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$, $u\in BV(\Omega)\cap L^{\infty}{(\Omega)}$ and let $\Sigma\subset\Omega$ be an oriented countably ${\mathcal{H}}^{N-1}$–rectifiable set. Then $u\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and the normal traces of $u\boldsymbol{A}$ on $\Sigma$ are given by

[TABLE]

3. Some remarks on $L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$

In this section we analyze the properties of the functional spaces needed to define the pairing $\left(\boldsymbol{A},Du\right)_{\lambda}$ introduced in (4).

Definition 3.1**.**

Given $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$, let us define the spaces:

[TABLE]

Notice that $|\operatorname{div}\boldsymbol{A}|\ll{\mathcal{H}}^{N-1}$ and $u^{*}$ is defined ${\mathcal{H}}^{N-1}$-a.e. in $\Omega$, hence the definitions are well-posed.

The following lemma shows that if $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ then any representative $u^{\lambda}$ of $u$ defined in (11) (in particular $u^{+}$, $u^{-}$) is summable with respect to the measure $|\operatorname{div}\boldsymbol{A}|$, hence the definitions of the spaces $BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ and $BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ are independent of the choice of the pointwise representative.

Lemma 3.2**.**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and let $u\in BV_{{\rm loc}}(\Omega)$. Given two Borel selections $\lambda,\mu\colon\Omega\to[0,1]$, then it holds:

• (i)

$u^{\lambda}\in L^{1}_{\rm{loc}}(\Omega,|\operatorname{div}\boldsymbol{A}|)$* if and only if $u^{\mu}\in L^{1}_{\rm{loc}}(\Omega,|\operatorname{div}\boldsymbol{A}|)$;*

• (ii)

for every countably ${\mathcal{H}}^{N-1}$-rectifiable set $\Sigma\subset\Omega$, $u^{\lambda}\in L^{1}_{\rm{loc}}(\Sigma,{\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Sigma)$ if and only if $u^{\mu}\in L^{1}_{\rm{loc}}(\Sigma,{\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Sigma)$.

Proof.

We prove only (i), being the proof of (ii) entirely similar. By the representation (14) of $\operatorname{div}\boldsymbol{A}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits J_{u}$ and the estimate (13), for every compact set $K\Subset\Omega$ we have

[TABLE]

Recalling that $u^{+}-u^{-}=0$ in $\Omega\setminus S_{u}$, i.e. ${\mathcal{H}}^{N-1}$-a.e. in $\Omega\setminus J_{u}$, it follows that $u^{+}-u^{-}\in L^{1}_{\rm{loc}}(\Omega,|\operatorname{div}\boldsymbol{A}|)$. The result now follows by observing that $u^{\lambda}=u^{\mu}+(\lambda-\mu)(u^{+}-u^{-}).$ ∎

We underline that, for every $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$ and every $u\in BV(\Omega)$, it holds

[TABLE]

Nevertheless, in general the functions $|\operatorname{Tr}^{i,e}(\boldsymbol{A},J_{u})|\,u^{\pm}$ are not summable with respect to ${\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits J_{u}$, even under the additional assumption $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$, as it is shown in the following example.

Example 3.3*.*

Let $\Omega=B_{1}(0)\subset\mathbb{R}^{2}$. Let us show that there exist a vector field $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$ and a function $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ such that

[TABLE]

Let $1=r_{0}>r_{1}>\cdots>r_{n}>\cdots$ be a decreasing sequence converging to [math], such that

[TABLE]

and let $u\colon\Omega\to\mathbb{R}$ be defined by $u(x)=j$, if $r_{j}\leq|x|<r_{j-1}$, $j\in\mathbb{N}$. Since

[TABLE]

then $u\in BV(\Omega)$. We choose on the jump set $J_{u}=\bigcup_{j=1}^{\infty}\partial B_{r_{j}}(0)$ the orientation such that $u^{i}=u^{+}=j+1$ and $u^{e}=u^{-}=j$ on $\partial B_{r_{j}}(0)$.

Let $(a_{j})\subset\mathbb{R}$ be a bounded sequence, and let

[TABLE]

We have that $\boldsymbol{A}\in L^{\infty}(\Omega,\mathbb{R}^{2})$, $\operatorname{Tr}^{i}(\boldsymbol{A},J_{u})=a_{j+1}$, $\operatorname{Tr}^{e}(\boldsymbol{A},J_{u})=a_{j}$ on $\partial B_{r_{j}}$, and

[TABLE]

so that $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$. On the other hand, if we choose a sequence $(a_{j})$ such that $|a_{j}|\geq c>0$ for every $j\in\mathbb{N}$, we have that

[TABLE]

We conclude this example observing that, with the choice $a_{j}=(-1)^{j}$, we have also that

[TABLE]

We collect here the main features of the truncation operator that will be useful to generalize to $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ properties valid in $BV(\Omega)\cap L^{\infty}{(\Omega)}$.

Proposition 3.4** (Properties of the truncated functions).**

For every $k>0$, let

[TABLE]

Let $u\in BV(\Omega)$ and let $\lambda\colon\Omega\to[0,1]$ be a Borel function. Then the following hold.

• (i)

$T_{k}(u^{\pm})=[T_{k}(u)]^{\pm}\to u^{\pm}$, $[T_{k}(u)]^{\lambda}\to u^{\lambda}$, ${\mathcal{H}}^{N-1}$-a.e. in $\Omega$;

• (ii)

$|DT_{k}(u)|\leq|Du|$* in the sense of measures, for every $k>0$;*

• (iii)

$|[T_{k}(u)]^{\pm}|\leq|u^{\pm}|$* for every $k>0$, hence*

[TABLE]

• (iv)

if $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$, then $T_{k}(u^{\lambda})\to u^{\lambda}$ in $L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$.

Proof.

The proof of (i) can be found in [AFP, Theorem 4.34(a)].

The inequality in (ii) is a consequence of the fact that $T_{k}$ is a $1$-Lipschitz function (see the first part of the proof of Theorem 3.96 in [AFP]).

The inequalities in (iii) follow from $|T_{k}(s)|\leq|s|$ and the equalities in (i), whereas (iv) follows from (iii), Lemma 3.2, and Lebesgue’s Dominated Convergence Theorem. ∎

4. Definition and basic properties of pairings

Definition 4.1** (Generalized pairing).**

Given a vector field $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and a Borel function $\lambda\colon\Omega\to[0,1]$, for every $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ the $\lambda$–pairing between $\boldsymbol{A}$ and $Du$ is the distribution $\left(\boldsymbol{A},Du\right)_{\lambda}\colon C^{\infty}_{c}(\Omega)\to\mathbb{R}$ acting as

[TABLE]

Remark 4.2*.*

The standard pairing

[TABLE]

introduced in [Anz], and deeply studied in recent years (see e.g. [ChenFrid], [CD3] and the references therein), is the $\lambda$-pairing corresponding to the constant selection $\lambda(x)=\frac{1}{2}$ for every $x\in\Omega$.

Remark 4.3*.*

The definition of generalized pairing and the properties proved in the rest of the paper can be extended straightforwardly to vector fields $\boldsymbol{A}\in\mathcal{DM}^{\infty}_{{\rm loc}}{(\Omega)}$ and functions $u\in BV_{\rm loc}({\Omega})\cap L^{1}_{\rm loc}(\Omega,|\operatorname{div}\boldsymbol{A}|)$.

Clearly, the change of pointwise values of $u$ may just affect the behavior of the pairing on the jump set $J_{u}$ of $u$. More precisely, the following basic properties hold.

Proposition 4.4**.**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$, $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$, and $\lambda\colon\Omega\to[0,1]$ be a Borel function. Then $\left(\boldsymbol{A},Du\right)_{\lambda}$ is a Radon measure in $\Omega$, and the equations

[TABLE]

[TABLE]

hold in the sense of measures in $\Omega$. Moreover, $\left(\boldsymbol{A},Du\right)_{\lambda}$ is absolutely continuous with respect to $|Du|$, and

[TABLE]

In what follows we will write

[TABLE]

where $\theta_{\lambda}(\boldsymbol{A},Du,\cdot)$ denotes the Radon–Nikodým derivative of $\left(\boldsymbol{A},Du\right)_{\lambda}$ with respect to $|Du|$.

Proof.

Assume, in addition, that $u\in BV(\Omega)\cap L^{\infty}{(\Omega)}$. In this case the fact that $\left(\boldsymbol{A},Du\right)_{\lambda}$ is a Radon measure, and the validity of (19) are straightforward consequences of the fact that the distribution

[TABLE]

is a Radon measure in $\Omega$ (see [ChenFrid]). Moreover, we have that

[TABLE]

From [SchSch, Proposition 3.5] (see in particular formula (3.9) there), we have that

[TABLE]

Since $\left(\boldsymbol{A},Du\right)_{\lambda}=(1-\lambda)\left(\boldsymbol{A},Du\right)_{0}+\lambda\left(\boldsymbol{A},Du\right)_{1}$, (21) follows.

Consider now the general case $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$. Let $u_{k}:=T_{k}(u)$ be the sequence of truncated functions. By Proposition 3.4(i) and (iv), we have that $(u_{k})^{\lambda}\to u^{\lambda}$ ${\mathcal{H}}^{N-1}$-a.e. in $\Omega$ and in $L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$. Hence, we can pass to the limit in

[TABLE]

and obtain that $\left(\boldsymbol{A},Du\right)_{\lambda}$ is the weak∗ limit of $\left(\boldsymbol{A},Du_{k}\right)_{\lambda}$ in the sense of measures, so that (19) follows. Since, by the estimate (21) and Proposition 3.4(ii), we have that

[TABLE]

we conclude that (19), (20) and (21) hold in the sense of measures. ∎

Remark 4.5*.*

In the last part of the proof of Proposition 4.4 we have shown that, for every $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and every $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$, the pairing $\left(\boldsymbol{A},Du\right)_{\lambda}$ is the weak∗ limit, in the sense of measures, of the sequence $\left(\boldsymbol{A},DT_{k}(u)\right)_{\lambda}$.

Remark 4.6*.*

Since $(u+v)^{+}\leq u^{+}+v^{+}$ and $(u+v)^{-}\geq u^{-}+v^{-}$, with possibly strict inequalities, the map $u\mapsto\left(\boldsymbol{A},Du\right)_{\lambda}$ is not linear, in general. On the other hand, the map $u\mapsto u^{*}$ is linear, hence the standard pairing is linear with respect to $u$. More precisely, the $\lambda$-pairing is linear if and only if $\left(\boldsymbol{A},Du\right)_{\lambda}=\left(\boldsymbol{A},Du\right)_{*}$ for every $u\in BV(\mathbb{R}^{N})\cap L^{\infty}{(\mathbb{R}^{N})}$. Indeed, for every $u\in BV(\mathbb{R}^{N})\cap L^{\infty}{(\mathbb{R}^{N})}$ we have that

[TABLE]

Hence, if there exists $u\in BV(\mathbb{R}^{N})\cap L^{\infty}{(\mathbb{R}^{N})}$ such that $\left(\boldsymbol{A},Du\right)_{\lambda}\neq\left(\boldsymbol{A},Du\right)_{*}$, then the claim follows from (20).

Using (20), and the results of Theorem 3.3 in [CD3], we are able to compute explicitly the diffuse part $\left(\boldsymbol{A},Du\right)_{\lambda}^{d}$, the absolutely continuous part $\left(\boldsymbol{A},Du\right)_{\lambda}^{a}$, and the jump part $\left(\boldsymbol{A},Du\right)_{\lambda}^{j}$ of the generalized pairing.

Proposition 4.7**.**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$. Then the diffuse, the absolutely continuous and the jump part of the measure $\left(\boldsymbol{A},Du\right)_{\lambda}$ are respectively

[TABLE]

where $\operatorname{Tr}^{i}(\boldsymbol{A},J_{u})$ and $\operatorname{Tr}^{e}(\boldsymbol{A},J_{u})$ are the normal traces corresponding to the orientation of $J_{u}$ such that $u^{+}={u^{i}}$.

Proof.

By (20), $\left(\boldsymbol{A},Du\right)_{\lambda}$ and $\left(\boldsymbol{A},Du\right)_{*}^{d}$ may differ only on $J_{u}$, hence $\left(\boldsymbol{A},Du\right)_{\lambda}^{d}=\left(\boldsymbol{A},Du\right)_{*}^{d}$. Moreover, by Theorem 3.2 in [ChenFrid], $\left(\boldsymbol{A},Du\right)_{\lambda}^{a}=\left(\boldsymbol{A},Du\right)_{*}^{a}=\boldsymbol{A}\cdot\nabla u\,{\mathcal{L}}^{N}$.

Concerning the jump part $\left(\boldsymbol{A},Du\right)_{\lambda}^{j}$, by (21), we already know that it is concentrated on $J_{u}$. Denoting by $\alpha^{i}:=\operatorname{Tr}^{i}(\boldsymbol{A},J_{u})$ and $\alpha^{e}:=\operatorname{Tr}^{e}(\boldsymbol{A},J_{u})$, by Theorem 3.3 in [CD3] we already know that

[TABLE]

Finally, by (20) and (14), we conclude that

[TABLE]

Remark 4.8* (The pairing trivializes on $W^{1,1}$).*

From Proposition 4.7, we have that

[TABLE]

Remark 4.9* ($BV$ vector fields).*

If $\boldsymbol{A}\in BV(\Omega,\mathbb{R}^{N})\cap L^{\infty}(\Omega,\mathbb{R}^{N})$, then clearly $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and

[TABLE]

where $\boldsymbol{A}^{i,e}_{J_{u}}$ are the traces of $\boldsymbol{A}$ on $J_{u}$ in the sense of $BV$ (see [AFP, Theorem 3.77]). Hence, the jump part of $\left(\boldsymbol{A},Du\right)_{\lambda}$ can be written as

[TABLE]

Remark 4.10* (The pairing trivializes for continuous vector fields).*

If $\boldsymbol{A}\in C(\Omega,\mathbb{R}^{N})$, then by [CD3, Theorem 3.3] and [ComiPayne, Theorem 3.7] it holds

[TABLE]

The following result is an improvement of Proposition 4.15 in [CD3], Theorem 1.2 in [ChenFrid] and Lemma 2.2 in [Anz].

Proposition 4.11** (Approximation by $C^{\infty}$ fields).**

Let $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$. Then there exists a sequence $(\boldsymbol{A}_{k})_{k}$ in $C^{\infty}(\Omega,\mathbb{R}^{N})\cap L^{\infty}(\Omega,\mathbb{R}^{N})$ satisfying the following properties.

• (i)

$\boldsymbol{A}_{k}-\boldsymbol{A}\to 0$* in $L^{1}(\Omega,\mathbb{R}^{N})$, $\int_{\Omega}|\operatorname{div}\boldsymbol{A}_{k}|\,dx\to|\operatorname{div}\boldsymbol{A}|(\Omega)$, and $(\boldsymbol{A}_{k})_{k}$ is uniformly bounded.*

• (ii)

$\operatorname{div}\boldsymbol{A}_{k}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\operatorname{div}\boldsymbol{A}$* in the weak*∗* sense of measures in $\Omega$.*

• (iii)

For every oriented countably ${\mathcal{H}}^{N-1}$-rectifiable set $\Sigma\subset\Omega$ it holds

[TABLE]

where $\operatorname{Tr}^{*}(\boldsymbol{A},\Sigma):=[\operatorname{Tr}^{i}(\boldsymbol{A},\Sigma)+\operatorname{Tr}^{e}(\boldsymbol{A},\Sigma)]/2$.

Moreover, for every $u\in BV(\Omega)\cap L^{\infty}{(\Omega)}$, it holds

• (iv)

$\left(\boldsymbol{A}_{k},Du\right)_{*}\stackrel{{\scriptstyle*}}{{\rightharpoonup}}\left(\boldsymbol{A},Du\right)_{*}$* locally in the weak*∗* sense of measures in $\Omega$;*

• (v)

the sequence $\theta(\boldsymbol{A}_{k},Du;\cdot)$ weakly∗* converges in $L^{\infty}(\Omega,|Du|)$ to $\theta(\boldsymbol{A},Du;\cdot)$, where $\theta(\boldsymbol{A},Du;\cdot)$ is the Radon–Nikodým derivative of $\left(\boldsymbol{A},Du\right)_{*}$ with respect to $|Du|$.*

Remark 4.12*.*

It is not difficult to show that a similar approximation result holds also for $\boldsymbol{A}\in\mathcal{DM}^{\infty}_{{\rm loc}}{(\Omega)}$ with a sequence $(\boldsymbol{A}_{k})$ in $C^{\infty}(\Omega,\mathbb{R}^{N})$.

Proof.

(i) This part is proved in [ChenFrid, Theorem 1.2]. We just recall, for later use, that for every $k$ the vector field $\boldsymbol{A}_{k}$ is of the form

[TABLE]

where $(\varphi_{i})$ is a partition of unity subordinate to a locally finite covering of $\Omega$ depending on $k$ and, for every $i$, $\varepsilon_{i}\in(0,1/k)$ is chosen in such a way that

[TABLE]

(see [ChenFrid], formula (1.8)).

(ii) From (i) we have that

[TABLE]

hence (ii) follows from the density of $C^{1}_{c}(\Omega)$ in $C_{0}(\Omega)$ in the norm of $L^{\infty}(\Omega)$ and the bound $\sup_{k}\int_{\Omega}|\operatorname{div}\boldsymbol{A}_{k}|\,dx<+\infty$.

(iii) As a first step we prove that, for every $u\in BV(\Omega)\cap L^{\infty}{(\Omega)}$,

[TABLE]

Specifically, from the definition (23) of $\boldsymbol{A}_{k}$ and the identity $\sum_{i}\nabla\varphi_{i}=0$ we have that

[TABLE]

From the estimate (24) we have that

[TABLE]

and hence, to prove (25), it is enough to show that

[TABLE]

On the other hand,

[TABLE]

hence (26) follows by observing that the functions $\rho_{\varepsilon_{i}}\ast(u\,\varphi)$ converge pointwise $\mathcal{H}^{N-1}$–a.e. in $\Omega$ to $u^{*}\varphi$, so that

[TABLE]

We remark that, as a consequence of (25), if $E\Subset\Omega$ is a set of finite perimeter, then

[TABLE]

Let us now prove (iii). Let $\omega\Subset\Omega$ be a set of class $C^{1}$. By the definition (12) of normal traces, by (i), (ii) and (27), for every $\varphi\in C^{\infty}_{c}(\Omega)$ we have that

[TABLE]

so that

[TABLE]

Hence, by (14), we have proved that

[TABLE]

in the sense of distributions. Using the arguments of Section 2.4, this relation can be extended to the countably ${\mathcal{H}}^{N-1}$-rectifiable set $\Sigma$. By a density argument as in (ii), this relation hold for every $\varphi\in C_{c}(\Omega)$, hence (iii) holds true for $\operatorname{Tr}^{e}(\boldsymbol{A}_{k},\Sigma)$. Finally, a similar computation holds for $\operatorname{Tr}^{i}(\boldsymbol{A}_{k},\Sigma)$.

(iv) Using the passage to the limit in (25) we obtain straightforwardly

[TABLE]

for every $\varphi\in C^{1}_{c}(\Omega)$. The validity of this relation for $\varphi\in C_{c}(\Omega)$ follows from (21) and the fact that the sequence $(\boldsymbol{A}_{k})$ is bounded in $L^{\infty}(\Omega,\mathbb{R}^{N})$.

(v) Using the definition (22) of the density $\theta$, we have that, for every $\varphi\in C_{c}(\Omega)$,

[TABLE]

Since, by (21) and (22), the sequence $(\theta(\boldsymbol{A}_{k},Du,\cdot))$ is bounded in $L^{\infty}(\Omega,|Du|)$, then (v) follows. ∎

5. Coarea formula for generalized pairings

This section is devoted to the proof of the coarea formula for the $\lambda$-pairing, and a related slicing result for its density $\theta_{\lambda}$.

Theorem 5.1** (Coarea formula).**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and let $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$. Then $\chi_{\{u>t\}}\in BV(\Omega)$ for ${\mathcal{L}}^{1}$-a.e. $t\in\mathbb{R}$, and

[TABLE]

Proof.

Since $\left(\boldsymbol{A},Du\right)_{\lambda}$ and $\left(\boldsymbol{A},D\chi_{\{u>t\}}\right)_{\lambda}$ are measures in $\Omega$ for ${\mathcal{L}}^{1}$-a.e. $t\in\mathbb{R}$, it is enough to prove (28) for $\varphi\in C^{\infty}_{c}(\Omega)$.

Let us first consider the case $u\in L^{\infty}(\Omega)$. By possibly replacing $u$ with $u+\|u\|_{\infty}$, it is not restrictive to assume that $u\geq 0$. Given a test function $\varphi\in C^{\infty}_{c}(\Omega)$, we have that

[TABLE]

Moreover, by [DCFV2, Lemma 2.2], we have that, for $\mathcal{L}^{1}$-a.e. $t\in\mathbb{R}$, there exists a Borel set $N_{t}\subset\Omega$, with ${\mathcal{H}}^{N-1}(N_{t})=0$, such that

[TABLE]

so that, since $|\operatorname{div}\boldsymbol{A}|\ll{\mathcal{H}}^{N-1}$, we obtain that

[TABLE]

Hence, we get

[TABLE]

As a consequence, from (29), (30) and the definition (18) of $\left(\boldsymbol{A},Du\right)_{\lambda}$, we conclude that (28) holds for every test function $\varphi\in C^{\infty}_{c}(\Omega)$ and for every $u\in BV(\mathbb{R}^{N})\cap L^{\infty}{(\mathbb{R}^{N})}$.

Finally, the general case $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ follows applying the previous step to the truncated functions $u_{k}:=T_{k}(u)$. Specifically, (28) gives, for every $k>0$,

[TABLE]

By Remark 4.5, the left-hand side of (31) converges to $\left\langle\left(\boldsymbol{A},Du\right)_{\lambda}\,,\,\varphi\right\rangle$. On the other hand, since

[TABLE]

the right-hand side in (31) is equal to

[TABLE]

By the estimate (21) we have that

[TABLE]

and hence, by the coarea formula in $BV$ and the Lebesgue Dominated Convergence Theorem, the integral in (32) converges to the right-hand side of (28) as $k\to+\infty$. ∎

Proposition 5.2**.**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and $u\in BV(\Omega)\cap L^{\infty}{(\Omega)}$. Then

[TABLE]

Proof.

Thanks to Proposition 4.11(iv), the proof can be done following the lines of [Anz, Proposition 2.7(iii)]. For the reader’s convenience, we recall here the main points.

Given two real numbers $a<b$, the function $v:=\max\{\min\{u,b\},a\}$ satisfies

[TABLE]

Since

[TABLE]

(see [GMS1, §4.1.4, Theorem 2(i)]), we deduce that

[TABLE]

Let $(\boldsymbol{A}_{k})\subset C^{\infty}(\Omega,\mathbb{R}^{N})\cap L^{\infty}(\Omega,\mathbb{R}^{N})$ be the sequence of smooth vector fields approximating $\boldsymbol{A}$ as in Proposition 4.11. Since, by [Anz, Proposition 2.3], we have

[TABLE]

then, from Proposition 4.11(v) and by the uniqueness of the limit in the $L^{\infty}(\Omega,|Dv|)$ weak∗ topology, we obtain that

[TABLE]

Recalling the definition (22) of $\theta_{\lambda}$ and the relation (20), we conclude that

[TABLE]

Specifically, $\theta_{\lambda}(\boldsymbol{A},Du,x)=\theta(\boldsymbol{A},Du,x)=\theta(\boldsymbol{A},Dv,x)=\theta_{\lambda}(\boldsymbol{A},Dv,x)$ for $|D^{d}v|$-a.e. $x\in\Omega$, whereas, by Proposition 4.7 (and using the notations therein) and the inclusion $J_{v}\subset J_{u}$, $\theta_{\lambda}(\boldsymbol{A},Du,x)=(1-\lambda)\operatorname{Tr}^{i}(\boldsymbol{A},J_{u})+\lambda\operatorname{Tr}^{e}(\boldsymbol{A},J_{u})=\theta_{\lambda}(\boldsymbol{A},Dv,x)$ for $|D^{j}v|$-a.e. $x\in\Omega$.

Given $\varphi\in C^{\infty}_{c}(\Omega)$, let us compute $\left\langle\left(\boldsymbol{A},Dv\right)_{\lambda}\,,\,\varphi\right\rangle$. By the definition of $\theta_{\lambda}(\boldsymbol{A},Dv,x)$, equality (35), the coarea formula in BV (see [AFP, Theorem 3.40]) and (34) it holds

[TABLE]

On the other hand, by the coarea formula (28) and (34), it holds

[TABLE]

Comparing (36) with (37), we finally conclude that, for every $a<b$,

[TABLE]

so that (33) follows. ∎

6. Chain rule, Leibniz and Gauss–Green formulas for generalized pairings

In this section we show that some relevant formulas, proved in [CD3] for the standard pairing, remain valid for general $\lambda$–pairings.

Proposition 6.1** (Chain Rule).**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and let $u\in BV(\Omega)\cap L^{\infty}{(\Omega)}$. Let $h:\mathbb{R}\to\mathbb{R}$ be a Lipschitz function. Then it holds:

• (i)

$\left(\boldsymbol{A},D[h(u)]\right)_{\lambda}^{d}=\left(\boldsymbol{A},D[h(u)]\right)_{*}^{d}$*, and $\left(\boldsymbol{A},D[h(u)]\right)_{\lambda}^{a}=h^{\prime}(\widetilde{u})\,\boldsymbol{A}\cdot\nabla u\,{\mathcal{L}}^{N}$. *

Moreover, if $h$ is non-decreasing, then

• (ii)

$\displaystyle\left(\boldsymbol{A},D[h(u)]\right)_{\lambda}^{j}=\frac{h(u^{+})-h(u^{-})}{u^{+}-u^{-}}\left(\boldsymbol{A},Du\right)_{\lambda}^{j};$**

• (iii)

$\theta_{\lambda}(\boldsymbol{A},D[h(u)],x)=\theta_{\lambda}(\boldsymbol{A},Du,x)$, for $|D[h(u)]|$-a.e. $x\in\Omega$.

The same characterization holds if $u\in BV_{{\rm loc}}(\Omega)\cap L^{\infty}_{{\rm loc}}{(\Omega)}$ and $h\colon I\to\mathbb{R}$ is a locally Lipschitz function such that $u(\Omega)\Subset I$.

Proof.

Although the proof is essentially the same of [CD3, Proposition 4.5], for the sake of completeness we prefer to illustrate it in some detail.

One of the main ingredients is the Chain Rule Formula for $BV$ functions (see [AFP, Theorem 3.99]):

[TABLE]

Statement (i) easily follows from the first two relations above and Proposition 4.7.

Concerning (ii), we have that $[h(u)]^{i,e}=h(u^{i,e})$ (see [AFP, Proposition 3.69(c)]). Moreover, since $h$ is non-decreasing, also the relations $[h(u)]^{\pm}=h(u^{\pm})$ hold true, and hence (ii) follows again from Proposition 4.7.

Let us prove (iii). If $h$ is strictly increasing, we can follow the proof of [Anz, Proposition 2.8]. Specifically, $\{u>t\}=\{h(u)>h(t)\}$ for every $t\in\mathbb{R}$, hence

[TABLE]

From Proposition 5.2, for $\mathcal{L}^{1}$-a.e. $t\in\mathbb{R}$ it holds

[TABLE]

for $|D\chi_{\{u>t\}}|$-a.e. $x\in\Omega$, and (iii) follows.

If $h$ is non-decreasing, we can adapt the proof of [LaSe, Proposition 2.7]. Specifically, let $h_{\varepsilon}(t):=h(t)+\varepsilon\,t$, so that $h_{\varepsilon}$ is strictly increasing for every $\varepsilon>0$. Since

[TABLE]

by the previous step we deduce that

[TABLE]

On the other hand,

[TABLE]

hence, passing to the limit in (38) as $\varepsilon\to 0$, we deduce that

[TABLE]

and (iii) follows. ∎

Proposition 6.2** (Leibniz formula).**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\Omega)}$ and $u,v\in BV(\Omega)\cap L^{\infty}{(\Omega)}$. Then, choosing on $J_{u}$ the orientation such that $u^{+}={u^{i}}$, it holds

[TABLE]

Proof.

By [CD3, Proposition 4.9], denoting $\alpha^{i}:=\operatorname{Tr}^{i}(\boldsymbol{A},J_{u})$ and $\alpha^{e}:=\operatorname{Tr}^{e}(\boldsymbol{A},J_{u})$ we have that

[TABLE]

hence (39) follows from (41) and Proposition 4.7.

From the representation formulas (14) and Proposition 2.6, we get

[TABLE]

hence, from (42), we obtain

[TABLE]

that is (40) holds. ∎

In the last part of this section we will prove a generalized Gauss–Green formula for vector fields $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\mathbb{R}^{N})}$ on a set $E\subset\mathbb{R}^{N}$ of finite perimeter, generalizing the analogous result for the standard pairing proved in [CD3, Theorem 5.1].

Using the conventions of Section 2.4, we will assume that the generalized normal vector on $\partial^{*}E$ coincides ${\mathcal{H}}^{N-1}$-a.e. on $\partial^{*}E$ with the measure–theoretic interior unit normal vector to $E$.

Theorem 6.3** (Gauss-Green).**

Let $\boldsymbol{A}\in\mathcal{DM}^{\infty}{(\mathbb{R}^{N})}$ and $u\in BV({\mathbb{R}^{N}})\cap L^{1}(\mathbb{R}^{N},|\operatorname{div}\boldsymbol{A}|)$. Let $E\subset\mathbb{R}^{N}$ be a bounded set with finite perimeter, and assume that the traces ${u^{e}},{u^{i}}$ of $u$ on $\partial^{*}E$ belong to $L^{1}(\partial^{*}E,{\mathcal{H}}^{N-1}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\partial^{*}E)$. Then the following Gauss–Green formulas hold:

[TABLE]

where $E^{1}$ is the measure theoretic interior of $E$ and $\partial^{*}E$ is oriented with respect to the interior unit normal vector.

Proof.

We recall that, by Lemma 3.2, $u^{\lambda}\in L^{1}_{\rm{loc}}(\mathbb{R}^{N},|\operatorname{div}\boldsymbol{A}|)$. Recalling (20), we have that

[TABLE]

On the other hand, by the definition (11) of $u^{\lambda}$, it holds

[TABLE]

so that (43) follows from the Gauss–Green formula for the standard pairing proved in [CD3, Theorem 5.1]. The validity of (44) can be checked in a very similar way. ∎

7. Semicontinuity results

In this section we consider the pairing as a function in $BV$

[TABLE]

where $\mathcal{M}_{b}(\Omega)$ denotes the space of finite Borel measures on $\Omega$ (see (21)).

Our aim is to characterize the selections $\lambda\colon\Omega\to[0,1]$ such that the above map is lower (resp. upper) semicontinuous, meaning that, if $(u_{n})\subset BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ is a sequence converging to a function $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ (in a suitable way), then

[TABLE]

Since $\left(\boldsymbol{A},Du\right)_{\lambda}$ is affected by the pointwise value of $u$, the correct notion of convergence in $BV$ seems to be the strict one (see e.g. [AFP, Definition 3.14]).

Definition 7.1**.**

The sequence $(u_{n})\subset BV(\Omega)$ strictly converges to $u\in BV(\Omega)$ if $(u_{n})$ converges to $u$ in $L^{1}(\Omega)$ and the total variations $|Du_{n}|(\Omega)$ converge to $|Du|(\Omega)$.

We recall a recent result concerning the pointwise behavior of strictly converging sequences.

Proposition 7.2**.**

Every sequence $(u_{n})$ strictly convergent in $BV(\Omega)$ to $u$ admits a subsequence $(u_{n_{k}})$ such that for ${\mathcal{H}}^{N-1}$-a.e. $x\in\Omega$

[TABLE]

In particular, $\lim_{k}\widetilde{u}_{n_{k}}(x)=\widetilde{u}(x)$ for ${\mathcal{H}}^{N-1}$-a.e. $x\in\Omega\setminus J_{u}$.

Proof.

See [La], Theorem 3.2, and Corollary 3.3. ∎

Combining Proposition 7.2 with Theorem 3.3 in [CDLP], we obtain the following approximation result.

Proposition 7.3**.**

Let $\Omega\subset\mathbb{R}^{N}$ be a bounded open set, and let $u\in BV(\Omega)$. Then there exist two sequences $(u_{n}),(v_{n})\subset W^{1,1}(\Omega)$ such that:

• (a)

for every $n\in\mathbb{N}$, $\widetilde{v}_{n}\leq u^{-}$ and $u^{+}\leq\widetilde{u}_{n}$ ${\mathcal{H}}^{N-1}$-a.e. in $\Omega$;

• (b)

$u_{n}\to u$, $v_{n}\to u$ strictly in $BV$;

• (c)

$\widetilde{u}_{n}(x)\to u^{+}(x)$* and $\widetilde{v}_{n}(x)\to u^{-}(x)$ for ${\mathcal{H}}^{N-1}$-a.e. $x\in\Omega$.*

If, in addition, $u\in L^{\infty}(\Omega)$, then the above sequences are bounded in $L^{\infty}(\Omega)$.

Proof.

From Theorem 3.3 in [CDLP], there exists a sequence $(u_{n})\subset W^{1,1}(\Omega)$, strictly convergent to $u$, and such that $\widetilde{u}_{n}\geq u^{+}$ ${\mathcal{H}}^{N-1}$-a.e. in $\Omega$, for every $n\in\mathbb{N}$. Moreover, if $u$ is bounded, then this sequence is bounded in $L^{\infty}(\Omega)$. By Proposition 7.2, we can extract a subsequence (not relabeled) such that

[TABLE]

On the other hand, the inequality $\widetilde{u}_{n}\geq u^{+}$ gives

[TABLE]

hence the assertion for $(u_{n})$ follows. The construction of $(v_{n})$ can be done in a similar way. ∎

In order to state the semicontinuity results, a more piece of notation is needed. Given a vector field $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$, let us denote by $\Omega_{\boldsymbol{A}}$ the set of points $x\in\Omega$ such that $x$ belongs to the support of $\operatorname{div}\boldsymbol{A}$ (i.e. $|\operatorname{div}\boldsymbol{A}|(B_{r}(x)\cap\Omega)>0$ for every $r>0$), and the limit

[TABLE]

exists in $\mathbb{R}$, with $|\psi_{\boldsymbol{A}}(x)|=1$. If we extend $\psi_{\boldsymbol{A}}=0$ in $\Omega\setminus\Omega_{\boldsymbol{A}}$, we have that $\psi_{\boldsymbol{A}}\in L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ and the polar decomposition $\operatorname{div}\boldsymbol{A}=\psi_{\boldsymbol{A}}|\operatorname{div}\boldsymbol{A}|$ holds. Moreover, if we define the sets

[TABLE]

then $(\operatorname{div}\boldsymbol{A})^{+}=\operatorname{div}\boldsymbol{A}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega^{+}_{\boldsymbol{A}}$ and $(\operatorname{div}\boldsymbol{A})^{-}=-\operatorname{div}\boldsymbol{A}\mathop{\hbox{\vrule height=7.0pt,width=0.5pt,depth=0.0pt\vrule height=0.5pt,width=6.0pt,depth=0.0pt}}\nolimits\Omega^{-}_{\boldsymbol{A}}$.

Let $\Theta_{\boldsymbol{A}}$ be the jump set of the measure $|\operatorname{div}\boldsymbol{A}|$ (see Proposition 2.3). Since $\Theta_{\boldsymbol{A}}$ is $\sigma$-finite with respect to ${\mathcal{H}}^{N-1}$, then there exists a countably ${\mathcal{H}}^{N-1}$-rectifiable Borel set $\Theta^{r}_{\boldsymbol{A}}\subseteq\Theta_{\boldsymbol{A}}$ such that $\Theta^{u}_{\boldsymbol{A}}:=\Theta_{\boldsymbol{A}}\setminus\Theta^{r}_{\boldsymbol{A}}$ is purely ${\mathcal{H}}^{N-1}$-unrectifiable (i.e. ${\mathcal{H}}^{N-1}(\Theta^{u}_{\boldsymbol{A}}\cap\Sigma)=0$ for every countably ${\mathcal{H}}^{N-1}$-rectifiable set $\Sigma$, see [AFP, Definition 2.64 and Proposition 2.76]).

Let us define the families of selections

[TABLE]

These families satisfy the following extremality properties.

Lemma 7.4**.**

Given $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$, $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$, $\varphi\in C_{0}(\Omega)$, $\varphi\geq 0$, then for every Borel function $\lambda\colon\Omega\to[0,1]$ it holds

[TABLE]

with equality if $\lambda\in\Lambda_{{\rm lsc}}$.

Similarly,

[TABLE]

with equality if $\lambda\in\Lambda_{{\rm usc}}$.

Proof.

Let us prove the claim only for the first inequality in (47), the other being similar.

Since, by the very definition of $\Omega^{+}_{\boldsymbol{A}}$,

[TABLE]

and $u^{\lambda}\leq u^{+}$ ${\mathcal{H}}^{N-1}$-a.e. in $\Omega$, the first inequality in (47) follows.

Let $\lambda\in\Lambda_{{\rm lsc}}$ and let us prove that equality holds in the first inequality in (47). Let us decompose the set $\Omega_{\boldsymbol{A}}^{+}$, defined in (46), as the union of the disjoint sets

[TABLE]

that, in turn, coincide up to sets of ${\mathcal{H}}^{N-1}$-measure zero respectively with

[TABLE]

Observe that $u^{\lambda}=\widetilde{u}$ ${\mathcal{H}}^{N-1}$-a.e. (hence $|\operatorname{div}\boldsymbol{A}|$-a.e.) in $\Omega_{\boldsymbol{A}}^{+}\setminus S_{u}$, $u^{\lambda}=u^{+}$ ${\mathcal{H}}^{N-1}$-a.e. in $\Omega_{\boldsymbol{A}}^{+}\cap\Theta^{r}_{\boldsymbol{A}}$, and, by Proposition 2.3, $|\operatorname{div}\boldsymbol{A}|((\Omega_{\boldsymbol{A}}^{+}\setminus\Theta_{\boldsymbol{A}})\cap J_{u})=0$ Hence,

[TABLE]

Corollary 7.5**.**

Given $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$ and $u\in BV({\Omega})\cap L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$, it holds:

[TABLE]

In particular,

[TABLE]

Moreover, if the orientation of $J_{u}$ is chosen in such a way that $u^{+}={u^{i}}$, then,

[TABLE]

Proof.

The first part is a direct consequence of the equality case in Lemma 7.4.

Let us prove (52). To simplify the notation, let

[TABLE]

Since $\left(\boldsymbol{A},Du\right)_{0}=-u^{-}\mu+\operatorname{div}(u\boldsymbol{A})$ and $\left(\boldsymbol{A},Du\right)_{1}=-u^{+}\mu+\operatorname{div}(u\boldsymbol{A})$, by definition of minimum of two measures, for every Borel set $E\subset\Omega$ one has

[TABLE]

where the infimum is taken over the pairs $E_{0},E_{1}$ of disjoint Borel sets such that $E=E_{0}\cup E_{1}$. Setting $E^{-}:=E\cap\Omega_{\boldsymbol{A}}^{-}$ and $E^{+}:=E\setminus E^{-}$, then $E\cap\Omega_{\boldsymbol{A}}^{+}\subset E^{+}$ and

[TABLE]

for every partition $\{E_{0},E_{1}\}$ of $E$. Hence,

[TABLE]

The proof of (53) is similar. Finally, (54) and (55) are consequences of (52) and (53), respectively, and Proposition 4.7. ∎

Theorem 7.6**.**

Let $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$, and let $\lambda\colon\Omega\to[0,1]$ be a Borel function.

Then $\lambda\in\Lambda_{{\rm lsc}}$ if and only if, for every $u_{n},u\in BV(\Omega)$ satisfying

• (a)

$u_{n}\to u$* strictly in $BV$,*

• (b)

there exists $g\in L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$ such that, for every $n\in\mathbb{N}$, $|u_{n}^{\pm}|\leq g$ $|\operatorname{div}\boldsymbol{A}|$-a.e. in $\Omega$,

it holds

[TABLE]

Analogously, $\lambda\in\Lambda_{{\rm usc}}$ if and only if, for every $u_{n},u\in BV(\Omega)$ satisfying (a), (b) it holds

[TABLE]

Proof.

Let us prove only the statement concerning the lower semicontinuity, the other being similar.

Let $\lambda\in\Lambda_{{\rm lsc}}$, let $u_{n},u\in BV(\Omega)$ satisfy (a), (b), and let us prove that the semicontinuity property in (56) holds. Let $\varphi\in C^{\infty}_{c}(\Omega)$, $\varphi\geq 0$, and let $(u_{n_{k}})$ be a subsequence such that

[TABLE]

and (45) holds true (here we use (a) and Proposition 7.2).

From Lemma 7.4, assumption (b), Fatou’s Lemma and the pointwise estimates (45) we have that

[TABLE]

Recalling that

[TABLE]

the same argument gives

[TABLE]

Since $|\operatorname{div}\boldsymbol{A}|(\Omega\setminus(\Omega_{\boldsymbol{A}}^{-}\cup\Omega_{\boldsymbol{A}}^{+}))=0$, from (58), (59) and the equality case in (47) we get

[TABLE]

Finally, from (60) and (a) we conclude that

[TABLE]

i.e., (56) holds true.

Assume now that (56) holds true for every $u_{n},u\in BV(\Omega)$ satisfying (a), (b), and let us prove that $\lambda\in\Lambda_{{\rm lsc}}$. We claim that, under these assumptions,

[TABLE]

in the sense of measures. By a truncation argument and Remark 4.5, it is enough to show that the above inequality holds for every $u\in BV(\Omega)\cap L^{\infty}(\Omega)$. Let $u\in BV(\Omega)\cap L^{\infty}(\Omega)$ and let $(u_{n}),(v_{n})\subset W^{1,1}(\Omega)\cap L^{\infty}(\Omega)$ be the approximating sequences given by Proposition 7.3. Observe that these sequences are bounded in $L^{\infty}(\Omega)$, so that they satisfy assumption (b). Since $(\widetilde{u}_{n})$ converges to $u^{+}$ $|\operatorname{div}\boldsymbol{A}|$-a.e. in $\Omega$ and, by (b), also in $L^{1}(\Omega,|\operatorname{div}\boldsymbol{A}|)$, for every test function $\varphi\in C^{\infty}_{c}(\Omega)$ we have that

[TABLE]

hence, by the semicontinuity assumption, if $\varphi\geq 0$,

[TABLE]

The same argument, using the sequence $(v_{n})$, shows that

[TABLE]

so that (61) follows.

Let $\Omega^{\prime}\Subset\Omega$ be an open domain with $C^{1}$ boundary. From Proposition 4.7 we have that

[TABLE]

hence, the inequalities (61) give

[TABLE]

Let $\Sigma\subset\Omega$ be an oriented countably ${\mathcal{H}}^{N-1}$-rectifiable set. Recalling the definition of normal traces given in Section 2.4, from (62) we deduce that

[TABLE]

Let us choose an orientation for the countably ${\mathcal{H}}^{N-1}$-rectifiable set $\Sigma^{+}:=\Theta_{\boldsymbol{A}}^{r}\cap\Omega_{\boldsymbol{A}}^{+}$. Since $\Sigma^{+}\subset\Omega_{\boldsymbol{A}}$ and $\psi_{\boldsymbol{A}}(x)=1$ for $|\operatorname{div}\boldsymbol{A}|$-a.e. $x\in\Sigma^{+}$, from (14) we have that

[TABLE]

Hence, from the first inequality in (63), we deduce that $\lambda=1$ ${\mathcal{H}}^{N-1}$-a.e. on $\Sigma^{+}$. A similar argument, using $\Sigma^{-}:=\Theta_{\boldsymbol{A}}^{r}\cap\Omega_{\boldsymbol{A}}^{-}$, shows that $\lambda=0$ ${\mathcal{H}}^{N-1}$-a.e. on $\Sigma^{-}$. ∎

Corollary 7.7**.**

Let $\boldsymbol{A}\in{\mathcal{DM}^{\infty}}(\Omega)$ and let $\lambda\colon\Omega\to[0,1]$ be a Borel function. Then the continuity property

[TABLE]

holds for every $u_{n},u\in BV(\Omega)$ satisfying (a) and (b) in Theorem 7.6 if and only if ${\mathcal{H}}^{N-1}(\Theta_{\boldsymbol{A}}^{r})=0$.

Proof.

We have that the stated property holds if and only if both (56) and (57) hold. From Theorem 7.6, these inequalities hold (for every $(u_{n})$, $u$) if and only if $\lambda\in\Lambda_{{\rm lsc}}\cap\Lambda_{{\rm usc}}$. Finally, from the very definition of $\Lambda_{{\rm lsc}}$ and $\Lambda_{{\rm usc}}$, we have that $\Lambda_{{\rm lsc}}\cap\Lambda_{{\rm usc}}\neq\emptyset$ if and only if ${\mathcal{H}}^{N-1}(\Theta_{\boldsymbol{A}}^{r})=0$. ∎

Remark 7.8*.*

The assumption ${\mathcal{H}}^{N-1}(\Theta_{\boldsymbol{A}}^{r})=0$ is trivially satisfied if $\operatorname{div}^{j}\boldsymbol{A}=0$, e.g. if $\operatorname{div}\boldsymbol{A}\in L^{1}(\Omega)$.

Example 7.9*.*

In view of Corollary 7.7 we have that, in general, the continuity property (64) does not hold with respect to the strict convergence in $BV$. Specifically, let $\Omega=(-2,2)\subset\mathbb{R}$ and consider $\boldsymbol{A}:=\chi_{(-1,1)}$, so that $\operatorname{div}\boldsymbol{A}=\delta_{-1}-\delta_{1}$, and $\Theta_{\boldsymbol{A}}^{r}=\Theta_{\boldsymbol{A}}=\{-1,+1\}$ is not empty. Let $\lambda\colon\Omega\to[0,1]$ be any Borel function. Let $u_{n}(x):=\max\{\min\{n+1-n|x|,1\},0\}$. It is readily seen that $(u_{n})$ strictly converges to $u:=\chi_{[-1,1]}$, so that $(-u_{n})$ strictly converges to $-u$, and $\left\langle\left(\boldsymbol{A},Du_{n}\right)_{\lambda}\,,\,\varphi\right\rangle=0$ for every $n$. On the other hand, choosing $\varphi$ such that $\varphi(-1)=0$ and $\varphi(1)=1$, one has

[TABLE]

and at least one of the right-hand sides must be different from [math].

Example 7.10*.*

We remark that, in general, (56) does not hold if assumption (a) is replaced by the weak∗ convergence in $BV$. Specifically, let us consider $\Omega=(-2,2)\subset\mathbb{R}$, $\boldsymbol{A}:=\chi_{(0,1)}$ and $u_{n}(x):=\max\{1-n|x|,0\}$. Since $(u_{n})$ converges to $u=0$ in $L^{1}(\Omega)$ and $|Du_{n}|(\Omega)=2$ for every $n$, then $(u_{n})$ converges weakly∗ to $u$ in $BV(\Omega)$. (see [AFP, Proposition 3.13]): If $\varphi\in C^{\infty}_{c}(\Omega)$ is strictly positive in [math], one has

[TABLE]

so that (56) does not hold.

Acknowledgments. The authors would like to thank Giovanni E. Comi for some useful remarks on a preliminary version of the manuscript. A.M. and V.D.C. have been partially supported by the Gruppo Nazionale per l’Analisi Matematica, la Probabilità e le loro Applicazioni (GNAMPA) of the Istituto Nazionale di Alta Matematica (INdAM). G.C. and A.M. have been partially supported by Sapienza - Ateneo 2017 Project ”Differential Models in Mathematical Physics”.