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 [G. Anzellotti,
Pairings between measures and bounded functions and compensated compactness,
Ann. Mat. Pura Appl. (4) 135 1983, 293–318], [G.-Q. Chen and H. Frid,
Divergence-measure fields and hyperbolic conservation laws,
Arch. Ration. Mech. Anal. 147 1999, 2, 89–118], 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
1 Introduction
In the seminal papers [6, 10], the product rule
for smooth functions u and regular vector fields
holds in the sense of measures.
The measure
The standard pairing turns out to be a basic tool
in many applications.
We mention here, among others:
extensions of the Gauss–Green formula
[6, 8, 9, 13, 14, 15, 16, 19, 18, 30];
the setting of the Euler–Lagrange equations associated
with integral functionals defined in
On the other hand, the standard pairing
is not adequate when dealing with
obstacle problems in
The main ingredients to build this family of pairings are the
absolute continuity of the measure
Recently, Scheven and Schmidt [34, 35, 36] have been in need to introduce the pairing
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
In this paper we prove that,
for every Borel function
where
We show that this freedom in the choice of
We characterize
the selections λ such that these functionals
are lower (resp. upper) semicontinuous with respect to
the strict convergence in
whereas the upper semicontinuity is characterized by
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
The plan of the paper is the following.
In Section 2 we recall some known results
on
The proofs of these results are based on
the analogous properties valid for the standard pairing
(see [19]), the fact that the generalized pairing
differs from the standard one only by a term concentrated
on
Our main application
of the above theory is proposed in Section 7,
where we consider the semicontinuity properties of the functionals
2 Notation and preliminary results
In the following Ω will always denote a nonempty open subset of
We denote by
If
Given an
of all points where E has density t.
The sets
Let
The function u is approximately continuous at
Given
In this case, x is a point of approximate
continuity, and
We say that
(2.1)
where
An
2.1 Measures
The space of all Radon measures on Ω will be denoted by
Given
for every μ-measurable set E and its positive and negative parts are defined, respectively, by
If
Given
We recall the following property (see [3, Proposition 2.56 and formula (2.41)]):
Given a nonnegative Borel
measure ν, we say that
We say that two positive measures
By the Radon–Nikodým theorem, given a nonnegative Radon measure ν, every
The following lemma shows the relation between the densities of μ and
Lemma 2.1.
Let
Proof.
Let
Moreover, from [3, Theorem 2.56 and (2.40)], the set
has zero
If we set
Taking
hence (2.3) follows because
Given
Proposition 2.2.
If
The set
is a Borel set, σ - finite with respect to
There exists
The measures
2.2 Functions of bounded variation
We say that
If
where
We denote by
At every point
Moreover, we can always choose an orientation on
Given a Borel function
When
Let E be an
We say that E is of finite perimeter in Ω if
Denoting by
If
exists in
A fundamental result of De Giorgi (see [3, Theorem 3.59]) states that
2.3 Divergence-measure fields
We will denote by
Similarly,
The basic properties of these vector fields are collected in the following proposition.
Proposition 2.3.
Let
2.4 Weak normal traces
In what follows, we
will deal with the traces of the normal component of a vector field
Given a domain
It turns out that this distribution is induced by an
Given a countably
for every
We can fix an orientation on Σ, given by
and the normal traces of
By a deep localization property proved in [1, Proposition 3.2],
these definitions are independent of the choice
of
We remark that the normal traces belong to
(see [1, Proposition 3.4]).
In particular, by (2.6),
Remark 2.4.
We observe that, if Σ is oriented by a normal vector field
ν and
so that the difference
The following result is a consequence of (2.7) and will be used in the study of the semicontinuity of the generalized pairing (see Theorem 7.6).
Theorem 2.5.
Let
Proof.
From (2.2) with
On the other hand, by (2.7)
and hence (2.8) holds.
Let us define the sets
By Proposition 2.3 (i) and [3, Theorems 2.22 and 2.83],
we infer that
For later use, we recall here a result proved in [19, Proposition 3.1].
Proposition 2.6.
Let
3 Some remarks on
L
1
(
Ω
,
|
div
𝑨
|
)
In this section we analyze the properties of the functional spaces
needed to define the pairing
Definition 3.1.
Given
Notice that
The following lemma shows that if
Lemma 3.2.
Let
For every countably
Proof.
We prove only (i), being the proof of (ii) entirely similar.
By the representation (2.7)
of
Recalling that
We underline that,
for every
Nevertheless, in general the functions
Example 3.3.
Let
Let
and let
it follows that
Let
We have that
so that
We conclude this example observing that, with the choice
We collect in the following proposition the main features of the truncation operator
that will be useful to generalize
to
Proposition 3.4 (Properties of the truncated functions).
For every
Let
If
Proof.
The proof of (i) can be found in [3, Theorem 4.34 (a)].
The inequality in (ii) is a consequence of the fact that
The inequalities in (iii) follow from
4 Definition and basic properties of pairings
Definition 4.1 (Generalized pairing).
Given a vector field
Remark 4.2.
The standard pairing
introduced in [6], and deeply studied in recent years (see e.g. [10], [19] and the references therein),
is the λ-pairing corresponding to the
constant selection
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
Clearly, the change of pointwise values of u may just affect the behavior of
the pairing on the jump set
Proposition 4.4.
Let
hold in the sense of measures in Ω.
Moreover,
In what follows we will write
where
Proof.
Assume, in addition, that
is a Radon measure in Ω (see [10]). Moreover, we have that
From [34, Proposition 3.5] (see in particular formula (3.9) there), we have that
Since
Consider now the general case
and obtain
that
we conclude that (4.2), (4.3) and (4.4) 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
Remark 4.6.
Since
Hence, if there exists
Using (4.3), and [19, results of Theorem 3.3],
we are able to
compute explicitly the diffuse part
Proposition 4.7.
Let
where
Proof.
By (4.3),
Concerning the jump part
Finally, by (4.3) and (2.7), we conclude that
Remark 4.8 (The pairing trivializes on
W
1
,
1
).
From Proposition 4.7, we have that
Remark 4.9 (BV vector fields).
If
where
Remark 4.10 (The pairing trivializes for continuous vector fields).
If
The following result is an improvement of [19, Proposition 4.15], [10, Theorem 1.2] and [6, Lemma 2.2].
Proposition 4.11 (Approximation by
C
∞
fields).
Let
For every oriented countably
where
Moreover, for every
The sequence
Remark 4.12.
It is not difficult to show that a similar approximation result holds also
for
Proof.
(i) This part is proved in [10, Theorem 1.2].
We just recall, for later use, that
for every k
the vector field
where
(see [10, formula (1.8)]).
(ii) From (i) we have that
hence statement (ii) follows from
the density of
(iii)
As a first step we prove that,
for every
Specifically, from the definition (4.6) of
From estimate (4.7) we have that
and hence, to prove (4.8), it is enough to show that
On the other hand,
hence (4.9) follows by observing that
the functions
We remark that, as a consequence of (4.8), if
Let us now prove (iii).
Let
so that
Hence, by (2.7), we have proved that
in the sense of distributions.
Using the arguments of Section 2.4, this relation
can be extended to the countably
(iv) Using the passage to the limit in (4.8) we obtain straightforwardly
for every
(v)
Using the definition (4.5) of the density θ,
we have that, for every
Since, by (4.4) and (4.5),
the sequence
5 Coarea formula for generalized pairings
This section is devoted to the proof of the coarea formula
for the λ-pairing,
and a related slicing result for its density
Theorem 5.1 (Coarea formula).
Let
Proof.
Since
Let us first consider the case
Moreover, by [20, Lemma 2.2], we have that,
for
so that, since
Hence, we get
As a consequence, from (5.2), (5.3) and
the definition (4.1) of
Finally, the general case
By Remark 4.5, the left-hand side of (5.4)
converges to
the right-hand side in (5.4) is equal to
By estimate (4.4) we have that
and hence, by the coarea formula in
Proposition 5.2.
Let
Proof.
Thanks to Proposition 4.11 (iv), the proof can be done following the lines of [6, Proposition 2.7 (iii)]. For the reader’s convenience, we recall here the main points.
Given two real numbers
(5.7)
Since
(see [25, Section 4.1.4, Theorem 2 (i)]), we deduce that
Let
from Proposition 4.11 (v)
and by the uniqueness of the limit in the
Recalling the definition (4.5) of
Specifically,
whereas, by Proposition 4.7
(and using the notations therein)
and the inclusion
Given
On the other hand, by the coarea formula (5.1) and (5.7), it holds
Comparing (5.9) with (5.10),
we finally conclude that, for every
so that (5.6) follows. ∎
6 Chain rule, Leibniz and Gauss–Green formulas for generalized pairings
In this section we show that some relevant formulas, proved in [19] for the standard pairing, remain valid for general λ-pairings.
Proposition 6.1 (Chain rule).
Let
Moreover, if h is non-decreasing, then:
The same characterization holds if
Proof.
Although the proof is essentially the same of [19, 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
Statement (i) easily follows from the first two relations above and Proposition 4.7.
Concerning (ii), we have that
Let us prove (iii). If h is strictly increasing, we can follow the proof of [6, Proposition 2.8]. Specifically,
From Proposition 5.2,
for
for
If h is non-decreasing, we can adapt the proof of
[29, Proposition 2.7].
Specifically, let
On the other hand,
hence, passing to the limit in (6.1) as
and (iii) follows. ∎
Proposition 6.2 (Leibniz formula).
Let
Proof.
By [19, Proposition 4.9],
denoting
hence (6.2) follows from (6.4) and Proposition 4.7.
From the representation formulas (2.7) and Proposition 2.6, we get
hence, from (6.5), we obtain
that is, (6.3) holds. ∎
In the last part of this section we will prove a generalized Gauss–Green formula
for vector fields
Using the conventions of Section 2.4,
we will assume that the generalized normal vector on
Theorem 6.3 (Gauss–Green).
Let
where
Proof.
We recall that, by Lemma 3.2,
On the other hand, by the definition (2.4) of
so that (6.6) follows from the Gauss–Green formula for the standard pairing proved in [19, Theorem 5.1]. The validity of (6.7) can be checked in a very similar way. ∎
7 Semicontinuity results
In this section we consider the pairing as a function in
where
Our aim is to characterize the selections
Since
Definition 7.1.
The sequence
We recall a recent result concerning the pointwise behavior of strictly converging sequences.
Proposition 7.2.
Every sequence
In particular,
Proof.
See [28, Theorem 3.2, and Corollary 3.3]. ∎
Combining Proposition 7.2 with [7, Theorem 3.3], we obtain the following approximation result.
Proposition 7.3.
Let
For every
If, in addition,
Proof.
From [7, Theorem 3.3], there exists a sequence
On the other hand, the inequality
hence the assertion for
In order to state the semicontinuity results, a more piece of notation is needed. To this end, given a vector field
exists in
then
Let
Let us define the families of selections
These families satisfy the following extremality properties.
Lemma 7.4.
Given
with equality if
Similarly,
with equality if
Proof.
Let us prove the claim only for the first inequality in (7.3), the other being similar.
Since, by the very definition of
and
Let
that, in turn, coincide up to sets of
Observe that
Corollary 7.5.
Given
In particular,
Moreover, if the orientation of
Proof.
The first part is a direct consequence of the equality case in Lemma 7.4.
Let us prove (7.4). To simplify the notation, let
Since
where the infimum is taken over the
pairs
for every partition
The proof of (7.5) is similar. Finally, (7.6) and (7.7) are consequences of (7.4) and (7.5), respectively, and Proposition 4.7. ∎
Theorem 7.6.
Let
there exists
it holds
Analogously,
Proof.
Let us prove only the statement concerning the lower semicontinuity, the other being similar.
Let
and (7.1) holds true (here we use (a) and Proposition 7.2).
From Lemma 7.4, assumption (b), Fatou’s Lemma and the pointwise estimates (7.1) we have that
Recalling that
the same argument gives
Since
Finally, from (7.12) and (a) we conclude that
i.e., (7.8) holds true.
Assume now that
(7.8) holds true
for every
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
hence, by the semicontinuity assumption, if
The same argument, using the sequence
so that (7.13) follows.
Let
hence, inequalities (7.13) give
Let
Let us choose an orientation for the countably
Hence,
from the first inequality in (7.15),
we deduce that
Corollary 7.7.
Let
holds for every
Proof.
We have that the stated property holds
if and only if
both (7.8) and (7.9) hold.
From Theorem 7.6,
these inequalities hold (for every
Remark 7.8.
The assumption
Example 7.9.
In view of Corollary 7.7 we have that,
in general,
the continuity property (7.16) does not hold
with respect to the strict convergence in
and at least one of the right-hand sides must be different from 0.
Example 7.10.
We remark that, in general,
(7.8) does not hold
if assumption (a) is replaced by
the weak
so that (7.8) does not hold.
Funding statement: Annalisa Malusa and Virginia De Cicco 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). Graziano Crasta and Annalisa Malusa have been partially supported by Sapienza – Ateneo 2017 Project “Differential Models in Mathematical Physics”.
Acknowledgements
The authors would like to thank Giovanni E. Comi for some useful remarks on a preliminary version of the manuscript.
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