A chain rule formula in BV and applications to conservation laws
aa r X i v : . [ m a t h . A P ] N ov A CHAIN RULE FORMULA IN BV AND APPLICATIONS TOCONSERVATION LAWS
GRAZIANO CRASTA ∗ AND
VIRGINIA DE CICCO † Abstract.
In this paper we prove a new chain rule formula for the distributional derivativeof the composite function v ( x ) = B ( x, u ( x )), where u :] a, b [ → R d has bounded variation, B ( x, · ) iscontinuously differentiable and B ( · , u ) has bounded variation. We propose an application of thisformula in order to deal in an intrinsic way with the discontinuous flux appearing in conservationlaws in one space variable. Key words.
Chain rule, BV functions, conservation laws with discontinuous flux AMS subject classifications.
Primary: 26A45, 35L65; Secondary: 26A24, 46F10
1. Introduction.
In 1967, A.I. Vol’pert in [23] (see also [24]), in view of ap-plications in the study of quasilinear hyperbolic equations, established a chain ruleformula for distributional derivatives of the composite function v ( x ) = B ( u ( x )) , where u : Ω → R has bounded variation in the open subset Ω of R N and B : R → R is con-tinuously differentiable. He proved that v has bounded variation and its distributionalderivative Dv (which is a Radon measure on Ω) admits an explicit representation interms of the gradient ∇ B and of the distributional derivative Du . More precisely,the following identity holds in the sense of measures: Dv = ∇ B ( u ) ∇ u L N + ∇ B ( e u ) D c u + [ B ( u + ) − B ( u − )] ν u H N − ⌊ J u , (1.1)where Du = ∇ u L N + D c u + ν u H N − ⌊ J u (1.2)is the usual decomposition of Du in its absolutely continuous part ∇ u with respectto the Lebesgue measure L N , its Cantor part D c u and its jumping part, which isrepresented by the restriction of the ( N − J u . Moreover, ν u denotes the measure theoretical unit normal to J u , e u isthe approximate limit and u + , u − are the approximate limits from both sides of J u .The validity of (1.1) is stated also in the vectorial case (see [2] and Theorem 3.96in [3]), namely if u : Ω → R d has bounded variation and B : R d → R is continuouslydifferentiable, then the terms in (1.1) should be interpreted in the following sense: Dv = ∇ B ( u ) · ∇ u L N + ∇ B ( e u ) · D c u + [ B ( u + ) − B ( u − )] ⊗ ν u H N − ⌊ J u . (1.3)The situation is significantly more complicated if B is only a Lipschitz continuousfunction. In this case, the general chain rule is false, while a weaker form of theformula was proved by Ambrosio and Dal Maso in [2] (see also [21]).On the other hand, in some recent papers a remarkable effort is devoted to es-tablish chain rule formulas with an explicit dependence on the space variable x .This amounts to describe the distributional derivative of the composite function v ( x ) = B ( x, u ( x )), where B ( x, · ) is continuously differentiable and, for every s ∈ R d , ∗ Dipartimento di Matematica “G. Castelnuovo”, Univ. di Roma I, P.le A. Moro 2, Roma, ItalyI-00185 ( [email protected]). † Dipartimento di Scienze di Base e Applicate per l’Ingegneria, Via A. Scarpa 10, Roma, ItalyI-00185 ( [email protected]) G. Crasta and V. De Cicco B ( · , s ) and u are functions with low regularity (which will be specified later). Theseformulas have applications, for example, in the study of the L lower semicontinuityof approximating linear integrals of convex non-autonomous functionals (see [11], [12]and [1]).The first formula of this type is established in [13] for functions u ∈ W , (Ω; R d )by assuming that, for every s ∈ R d , B ( · , s ) is an L function whose distributionaldivergence belongs to L (in particular it holds if B ( · , s ) ∈ W , (Ω; R d ) ).In [12] the formula is proved by assuming that, for every s ∈ R d , B ( · , s ) is an L function whose distributional divergence is a Radon measure with bounded totalvariation and u ∈ W , (Ω; R ) .The case of a function u ∈ BV (Ω) is studied in the papers [11] and [12]. In thefirst paper the authors have established the validity of the chain rule by requiringthat B ( · , s ) is differentiable in the weak sense for every s ∈ R . In the second one it isassumed only a BV dependence of B with respect to the variable x .The main difficulty of these results consists in giving sense to the different termsof the formula. Notice that the new term of derivation with respect to x needs aparticular attention. For instance in [12] this term is described by a Fubini’s typeinversion of integration order.The aim of this paper is to establish a chain rule formula for the distributionalderivative of the composite function v ( x ) = B ( x, u ( x )) , where u :] a, b [ → R d hasbounded variation, B ( x, · ) is continuously differentiable and B ( · , s ) has bounded vari-ation. We assume that there exists a countable set N ⊂ ] a, b [ such that the jump set J B ( · ,s ) of B ( · , s ) is contained in N for every s ∈ R d . Moreover we require that thereexists a positive finite Cantor measure λ on ] a, b [ such that ( D cx B )( · , s ) ≪ λ for every s ∈ R d . For every s ∈ R d let ψ ( · , s ) denote the Radon-Nikod´ym derivative of themeasure ( D cx B )( · , s ) with respect to λ , i.e. ψ ( · , s ) := d ( D cx B )( · , s ) dλ . We show that (see Theorem 4.1 below), under suitable additional assumptions, thecomposite function v ( x ) := B ( x, u ( x )) belongs to BV (] a, b [) and for any φ ∈ C (] a, b [)we have Z ] a,b [ φ ′ ( x ) v ( x ) dx = − Z ] a,b [ φ ( x )( ∇ x B )( x, u ( x )) dx − Z ] a,b [ φ ( x ) ψ ( x, u ( x )) dλ − Z ] a,b [ φ ( x )( D s B )( x, u ( x )) · ∇ u ( x ) dx − Z ] a,b [ φ ( x )( D s B )( x, u ( x )) · dD c u ( x ) − X x ∈N ∪ J u φ ( x ) [ B ( x + , u ( x + )) − B ( x − , u ( x − ))] , (1.4)where u ( x + ), u ( x − ) and B ( x + , s ), B ( x + , s ) are respectively the right and left limitsof u and B ( · , s ) at x .The proof is based on a regularization argument via convolutions and on theAmbrosio-Dal Maso derivation formula (see [2]). In order to prove the convergence chain rule formula in BV x , which requires a different nontrivial analysis dueto the possible interaction of the jump points of u and the jump points of B ( · , s ).In order to understand this effect, we consider firstly a piecewice constant function u , and we show that, in this case, the contributions of the jump parts can be collectedas in the summation in (1.4). The general case can be obtained by using a preciseapproximation result, proven in Section 3, of a BV function by piecewise constantfunctions which holds only for functions defined on an interval. By the way, weremark that this is one of the technical point where it is crucial the restriction to aone dimensional space variable.In Section 5, we consider the case d = 1 and we compare our chain rule withthe formula proven in [12]. We verify the (necessary!) coincidence of the terms ofderivation with respect to x in the case of piecewise constant functions u . Anyway,we remark that the form (1.4) is new also in this one-dimensional case.Finally, in Section 6 we discuss the use of our chain rule formula to conservationlaws with a discontinuous flux. The case of discontinuous fluxes has been intensivelystudied in the last few years (see e.g. [4, 5, 6, 7, 8, 9, 16, 18, 19, 20, 17, 22] and thereferences therein) due to a large class of applications in physical and traffic models.We do not address directly the issue of existence or uniqueness of solutions, forwhich we refer to the references listed above. We remark that the existence resultsare proved only for very special fluxes (tipically, only one jump in the space variable isallowed). For what concerns uniqueness, we recall a fairly general result by Audusseand Perthame [4], which is based on an extension of the classical Kruzkov method.In this framework, using our chain rule formula, we propose a definition of entropicsolution which is a generalization of the classical one valid for smooth fluxes (see e.g.[10]).We show that our definition is equivalent, under suitable assumptions, to thenotion of Kruzkov–type entropic solution obtained using the adapted entropies intro-duced by Audusse and Perthame in [4]. Our formula provides a neat environmentfor the treatment of all terms containing a derivative of the composition with a BV function which are present in equations of this type.We are inclined to believe that the methods here introduced can be useful to treatanalogous problems in the same context. Acknowledgements.
The authors would like to thank Gianni Dal Maso andNicola Fusco for stimulating discussions and suggestions during the preparation of themanuscript.
2. BV functions of one variable.
In this section we introduce the BV func-tions of one variable and we recall the definitions and the basic results (see the book[3] for a general survey on this subject).We recall that a function u = ( u , . . . , u d ) ∈ L (] a, b [; R d ) belongs to the space BV (] a, b [; R d ) if and only if T V ( u ) := sup n d X i =1 Z ba u i Dφ i dx : φ ∈ C (] a, b [; R d ) , k φ k ∞ ≤ o < + ∞ (2.1)(if d = 1 the usual notation is BV (] a, b [) ). This implies that the distributional deriva-tive D u = ( Du , . . . , Du d ) is a bounded Radon measure in ] a, b [ and the following G. Crasta and V. De Cicco integration by parts formula holds: Z ba u i Dφ i dx = − Z ba φ i dDu i ∀ φ ∈ C (] a, b [; R d ) , i = 1 , . . . , d . (2.2)A measure µ is absolutely continuous with respect to a positive measure λ ( µ ≪ λ insymbols) if µ ( B ) = 0 for every measurable set B such that λ ( B ) = 0 . We will oftenconsider the Lebesgue decomposition D u = ∇ u dx + D s u , (2.3)where ∇ u denotes the density of the absolutely continuous part of D u with respectto the Lebesgue measure on ] a, b [ , while D s u is its singular part .For every function u ∈ BV (] a, b [; R d ) the following left and right limits u ( x − ) := lim ε → ε Z xx − ε u ( y ) dy , u ( x + ) := lim ε → ε Z x + εx u ( y ) dy (2.4)exist at every point x ∈ ] a, b [ . In fact, u ( x − ) is well defined also in x = b , while u ( x + )exists also in x = a . The left and right limits just defined coincide a.e. with u andare left and right continuous, respectively.It is well known that the jump set of u , defined by J u := { x ∈ ] a, b [: u ( x − ) = u ( x + ) } is at most countable. The singular part D s u of the measure D u can be splitted intothe sum of a measure concentrated on J u and a measure D c u , called the Cantor part of D u , as in the following formula: D s u = D c u + (cid:0) u ( x + ) − u ( x − ) (cid:1) H ⌊ J u , (2.5)where H stands for the counting measure. Moreover, we consider the so-called diffusepart of the measure D u concentrated on C u :=] a, b [ \ J u and defined by e D u := ∇ u dx + D c u , (2.6)while D j u := (cid:0) u ( x + ) − u ( x − ) (cid:1) H ⌊ J u (2.7)is called the atomic part of D u . Analogously, we said that a nonnegative Borelmeasure µ is a Cantor measure if µ is a diffuse measure orthogonal to the Lebesguemeasure.If | D u | denotes the total variation measure of D u , we have that | D u | (] a, b [) equalsthe value of the supremum in (2.1); moreover, for every Borel subset B of ] a, b [, | D u | ( B ) = Z B |∇ u | ( x ) dx + | D c u | ( B ) + X x ∈ J u ∩ B | u ( x + ) − u ( x − ) | . (2.8)Now we recall the classical definition for BV functions of one variable, by meansof the pointwise variation ; for every function u :] a, b [ → R d , it is defined by pV ( u ) := sup n n − X i =1 | u ( t i +1 ) − u ( t i ) | : a < t < · · · < t n < b o . (2.9) chain rule formula in BV u having finite pointwise variation belongs to the space L ∞ (] a, b [; R d ), since its oscillation is controlled by pV ( u ). Moreover every boundedmonotone real valued function has finite pointwise variation and any (real valued)function having finite pointwise variation can be splitted into the difference of twomonotone functions.In order to avoid that u changes if it is modified even at a single point, weintroduced the following definition of essential variation eV ( u ) := sup n pV ( v ) : v = u a . e . in ] a, b [ o . (2.10)Finally, by Theorem 3.27 in [3], the essential variation eV ( u ) coincides with thevariation V ( u ), defined in (2.1). Any function u in the equivalence class of u (that is u = u a.e.) such that pV ( u ) = eV ( u ) = T V ( u ) is called a good representative . ByTheorem 3.28 in [3], we have that u is a good representative if and only if for every x ∈ ] a, b [ u ( x ) ∈ n θ u ( x − ) + (1 − θ ) u ( x + ) : θ ∈ [0 , o . (2.11)In particular, if (2.11) holds with θ = 0 (resp. θ = 1) for every x ∈ ] a, b [, we havethat u = u + (resp. u = u + ), while for θ = 1 / u coincides with the so-called preciserepresentative u ∗ ( x ) := u ( x + ) + u ( x − )2 . (2.12)Any good representative u is continuous in ] a, b [ \ J u , and it has a jump discontinuityat any point of J u satisfying u ( x − ) = u ( x − ), u ( x + ) = u ( x + ). Finally, any goodrepresentative u is a.e. differentiable in ] a, b [ and its derivative ∇ u coincides with thedensity of D u with respect to the Lebesgue measure. If not otherwise stated, in thispaper we always consider good representatives of BV functions.For every scalar BV function u the following coarea formula holds (see [14], The-orem 4.5.9): Z ba g ( x ) d | Du | ( x ) = Z + ∞−∞ dt Z { u ( x − ) ≤ t ≤ u ( x + ) } g ( x ) d H ( x ) (2.13)for every Borel function g :] a, b [ → [0 , + ∞ [.We remark that a Leibnitz rule formula in BV (] a, b [) holds: if v, w ∈ BV (] a, b [),then vw ∈ BV (] a, b [) and D ( vw ) = v ∗ Dw + w ∗ Dv, (2.14)in the sense of measures (see Example 3.97 in [3] and Remark 3.3 in [12]).Now we recall the properties of the convolution of a BV function. Let ϕ be astandard convolution kernel and let ( ϕ ε ) ε> be a family of mollifiers, i.e. ϕ ε ( x ) := ε − ϕ ( x/ε ). For every function u ∈ BV (] a, b [; R d ) we define u ε ( x ) := ( u ∗ ϕ ε )( x ) = Z ba ϕ ε ( x − y ) u ( y ) dy for x ∈ ] a ′ , b ′ [ ⊂⊂ ] a, b [ and 0 < ε < min( b − b ′ , a ′ − a ). We have that the mollifiedfunctions u ∗ ϕ ε converge a.e. to u in ] a, b [ and everywhere in [ a, b [ to the precise G. Crasta and V. De Cicco representative u ∗ (see Proposition 3.64(b) and Corollary 3.80 in [3]). Moreover ∇ u ε = ∇ ( u ∗ ϕ ε ) = ( D u ) ∗ ϕ ε (see Proposition 3.2 in [3]), where for a Radon measure µ , theconvolution µ ∗ ϕ ε is defined as( µ ∗ ϕ ε )( x ) := Z ba ϕ ε ( x − y ) dµ ( y ) . Finally, we recall that the measures ∇ u ε dx locally weakly ∗ converge in ] a, b [ to themeasure D u , i.e. for every φ ∈ C (] a, b [) we have Z ba φ ∇ u ε dx → Z ba φ dD u , as ε →
3. An approximation result.
In this section we exhibit an explicit piecewiseconstant approximation of a BV function, which is taylored to our needs in the proofof Theorem 4.1 . Lemma 3.1.
Let v ∈ BV (] a, b [) , let J denote its jump set, and let P ⊂ ] a, b [ \ J be a countable set. Then, for every ε > and every finite set P ε ⊂ P there exists apiecewise constant function v ε : ] a, b [ → R such that:(i) the (finite) jump set J ε of v ε contains all jumps of v of size greater than ε/ ;(ii) T V ( v ε ) ≤ T V ( v ) ;(iii) J ε ∩ P = ∅ and v ε ( x ) = v ( x ) for every x ∈ P ε ;(iv) v ε ( x + ) = v ( x + ) , v ε ( x − ) = v ( x − ) , for every x ∈ J ∩ J ε ;(v) | v ε ( x ) − v ( x ) | < ε for every x ∈ ] a, b [ \ J (the inequality holds everywhere if v is a good representative).Proof . Without loss of generality we can assume that v is a good representative.Let J = { x j } be the jump set of v . Since v ∈ BV , there exists N ∈ N such that X j>N | v ( x j + ) − v ( x j − ) | ≤ ε . (3.1)Let us define the functions v B , v S : ] a, b [ → R by v B ( x ) := P x j 6∈ { x , . . . , x N } ,v B ( x j − ) + v ( x j ) − v ( x j − ) , if x = x j for some j ≤ N,v S ( x ) := P x j 6∈ { x i : i > N } ,v S ( x j − ) + v ( x j ) − v ( x j − ) , if x = x j for some j > N. It is clear from the definition that the functions v B and v S take into account the bigand the small jumps of v respectively, and that the function v C := v − v B − v S iscontinuous in ] a, b [. In addition, v C is uniformly continuous in ] a, b [, since it can becontinuously extended to [ a, b ]. Then there exists δ > | v C ( x ) − v C ( y ) | < ε ∀ x, y ∈ ] a, b [ , | x − y | < δ. (3.2)Moreover, from (3.1) we have that | v S ( x ) | ≤ X j>N | v ( x j + ) − v ( x j − ) | < ε , ∀ x ∈ ] a, b [ . (3.3) chain rule formula in BV J ε = { y j } mj =0 , with a = y < y < · · · < y m = b , be a partition of [ a, b ]satisfying the following properties:(a) y i − y i − < δ for every i ∈ { , . . . , m } ;(b) x j ∈ J ε for every j ∈ { , . . . , N } ;(c) for every i ∈ { , . . . , m − } , if y i = x j for some j ∈ { , . . . , N } , then y i − , y i +1 6∈ { x , . . . , x N } ;(d) P ∩ J ε = ∅ ; moreover, each interval ] y i − , y i [ contains at most one point of P ε and, in that case, y i − , y i 6∈ { x , . . . , x N } .Let i ∈ { , . . . , m } . For every x, y ∈ ] y i − , y i [ we have that | v ( x ) − v ( y ) | ≤ | v ( x ) − v C ( x ) − v B ( x ) | + | v ( y ) − v C ( y ) − v B ( y ) | + | v C ( x ) − v C ( y ) | + | v B ( x ) − v B ( y ) | . Since ] y i − , y i [ does not contain points of { x , . . . , x N } we have that v B ( x ) = v B ( y ).Moreover, v − v C − v B = v S , hence by (3.2) and (3.3) we obtain | v ( x ) − v ( y ) | ≤ | v S ( x ) | + | v S ( y ) | + | v C ( x ) − v C ( y ) | < ε (3.4)( x, y ∈ ] y i − , y i [, i = 1 , . . . , m ).Finally, let us define the function v ε : ] a, b [ → R by v ε ( y i ) = v ( y i ) for every i ∈{ , . . . , m } , and, on every interval ] y i − , y i [ ( i ∈ { , . . . , m } ) by v ε ( x ) := v ( x ) , if ∅ 6 = P ε ∩ ] y i − , y i [= { x } ,v ( y i − ) , if y i ∈ { x , . . . , x N } ,v ( y i − ) , otherwise . It is clear from the construction that (i)–(iv) hold. Moreover, on every interval] y i − , y i [ ( i ∈ { , . . . , m } ) we have that v ε ( y i − ) = v ( y i − ) or v ε ( y i + ) = v ( y i + ) or v ε ( x ) = v ( x ) for some x ∈ ] y i − , y i [, hence from (3.4) we conclude that also (v) holds. Lemma 3.2. Let u ∈ BV (] a, b [; R d ) be a good representative, let J denote its jumpset, and let P ⊂ ] a, b [ \ J be a countable set. Then there exists a sequence of piecewiseconstant functions u n ∈ BV (] a, b [ , R d ) , n ∈ N , satisfying the following properties:(i) the (finite) jump set J n of u n does not contain points of P and contains alljumps x ∈ J such that | u ( x + ) − u ( x − ) | > /n ;(ii) T V ( u n ) ≤ T V ( u ) ;(iii) | u n ( x ) − u ( x ) | < C/n for every x ∈ ] a, b [ and n ∈ N , where C = 3 √ d ;(iv) for every x ∈ P there exists n x ∈ N such that u n ( x ) = u ( x ) for every n ≥ n x ;(v) for every x ∈ J there exists n x ∈ N such that u n ( x + ) = u ( x + ) , u n ( x − ) = u ( x − ) for every n ≥ n x .Proof . Let P = { z j } j . For every n ∈ N let us apply Lemma 3.1 to each component u i , i = 1 , . . . , d with ε = 3 /n and P ε = { z , . . . , z n } . The conclusion follows from thefact that J = S di =1 J u i and J n = S di =1 J u in , n ∈ N . 4. A chain rule formula in BV (] a, b [; R d ) . Let B :] a, b [ × R d → R be a functionsuch that B ( · , w ) ∈ BV (] a, b [) for all w ∈ R d . We recall that for every w ∈ R d ( D x B )( · , w ) = ( ∇ x B )( · , w ) dx + ( D cx B )( · , w ) + X x ∈N w [ B ( x + , w ) − B ( x − , w )] δ x (4.1)is the usual decomposition of the measure ( D x B )( · , w ) with respect to the Lebesguemeasure, where N w := J B ( · , w ) is the jump set of B ( · , w ) . Theorem 4.1. Let B :] a, b [ × R d → R be a locally bounded function such that G. Crasta and V. De Cicco (A1) for all w ∈ R d the function B ( · , w ) belongs to BV (] a, b [) and there exists acountable set N ⊂ ] a, b [ such that for every w ∈ R d we have N w ⊆ N ; (A2) for every compact set M ⊆ R d there exists a finite positive Borel measure µ M in ] a, b [ such that for every w , w ′ ∈ M and every Borel set A ⊆ ] a, b [ | ( D x B )( · , w ) − ( D x B )( · , w ′ ) | ( A ) ≤ | w − w ′ | µ M ( A ) ; (A3) for all x ∈ ] a, b [ \N the function B ( x, · ) belongs to C ( R d ) and, for everycompact set M ⊂ R d , there exists a constant D M > such that | ( D w B )( x, w ) | ≤ D M , ∀ x ∈ ] a, b [ \N , w ∈ M ; (A4) the function ( D w B ) ( · , w ) belongs to BV (] a, b [; R d ) for every w ∈ R d ;(A5) there exists a positive finite Cantor measure λ on ] a, b [ such that ( D cx B )( · , w ) ≪ λ for every w ∈ R d .Then for every u ∈ BV (] a, b [; R d ) the composite function v ( x ) := B ( x, u ( x )) , x ∈ ] a, b [ , belongs to BV (] a, b [) and for any φ ∈ C (] a, b [) we have Z ] a,b [ φ ′ ( x ) v ( x ) dx = − Z ] a,b [ φ ( x )( ∇ x B )( x, u ( x )) dx − Z ] a,b [ φ ( x ) ψ ( x, u ( x )) dλ − Z ] a,b [ φ ( x )( D w B )( x, u ( x )) · ∇ u ( x ) dx − Z ] a,b [ φ ( x )( D w B )( x, u ( x )) · dD c u ( x ) − X x ∈N ∪ J u φ ( x ) [ B ( x + , u ( x + )) − B ( x − , u ( x − ))] , (4.2) where for every w ∈ R d the function ψ ( · , w ) is the Radon-Nikod´ym derivative of themeasure ( D cx B )( · , w ) with respect to λ , i.e. ψ ( · , w ) := d ( D cx B )( · , w ) dλ . Remark 4.2. By ( A we obtain that for every compact set M ⊆ R d there existsa constant C M such that | ( D x B )( · , w ) | (] a, b [) ≤ C M , ∀ w ∈ M . (4.3) Moreover, for a.e. x ∈ ] a, b [ we have that | ( ∇ x B )( x, w ) | ≤ C M , ∀ w ∈ M , (4.4) and, for every x ∈ ] a, b [ , | ψ ( x, w ) | ≤ C M , ∀ w ∈ M . (4.5) chain rule formula in BV In addition, for a.e. x ∈ ] a, b [ and for every Borel set A ⊆ ] a, b [ the functions w ( ∇ x B )( x, w ) and w ( D cx B )( · , w )( A ) are Lipschitz continuous in R d . Finally, for λ -a.e. x ∈ ] a, b [ the function w ψ ( x, w ) is Lipschitz continuous too. Remark 4.3. Formula ( ) can be rewritten in a more explicit way as Z ] a,b [ φ ′ ( x ) v ( x ) dx = − Z ] a,b [ φ ( x )( ∇ x B )( x, u ( x )) dx − Z ] a,b [ φ ( x )( D w B )( x, u ( x )) · ∇ u ( x ) dx − Z ] a,b [ φ ( x ) ψ ( x, u ( x )) dλ − Z ] a,b [ φ ( x )( D w B )( x, u ( x )) · dD c u ( x ) − X x ∈N φ ( x ) (cid:20) B ( x + , u ( x + )) + B ( x + , u ( x − ))2 − B ( x − , u ( x + )) + B ( x − , u ( x − ))2 (cid:21) − X x ∈ J u φ ( x ) [ B ∗ ( x, u ( x + )) − B ∗ ( x, u ( x − ))] , (4.6) where for every x ∈ ] a, b [ and w ∈ R d B ∗ ( x, w ) := B ( x + , w ) + B ( x − , w )2 is the precise representative of the BV function x B ( x, w ) .In fact, it is easy to check that for every x ∈ J u ∩ N we have B ( x + , u ( x + )) − B ( x − , u ( x − )) = [ B ∗ ( x, u ( x + )) − B ∗ ( x, u ( x − ))]+ B ( x + , u ( x + )) + B ( x + , u ( x − ))2 − B ( x − , u ( x + )) + B ( x − , u ( x − ))2 ; in particular, for every x ∈ J u \ N we have B ( x + , u ( x + )) − B ( x − , u ( x − )) = B ( x, u ( x + )) − B ( x, u ( x − ))= B ∗ ( x, u ( x + )) − B ∗ ( x, u ( x − )) , and for every x ∈ N \ J u we have B ( x + , u ( x + )) − B ( x − , u ( x − )) = B ( x + , u ( x )) − B ( x − , u ( x ))= B ( x + , u ( x + )) + B ( x + , u ( x − ))2 − B ( x − , u ( x + )) + B ( x − , u ( x − ))2 . Proof of Theorem 4.1 Since the proof of Theorem 4.1 is rather long, it will be convenient to divide itinto several steps.In Step 1, following the regularization argument of Ambrosio–Dal Maso (see [2]),we consider the mollification B ε ( x, w ) of B ( x, w ) with respect to the first variable.We observe that, for every test function φ ∈ C c (] a, b [), the integral Z ] a,b [ φ ′ ( x ) B ε ( x, u ( x )) dx G. Crasta and V. De Cicco converges to the left-hand side of (1.4) as ε → + (see (4.8)). Then, for ε smallenough, we decompose this integral (using the chain rule formula for C functions) as − Z ] a,b [ φ ′ ( x ) B ε ( x, u ( x )) dx = Z ] a,b [ φ ( x ) D w B ε ( x, u ( x )) d ˜ D u ( x )+ X x ∈ J u φ ( x ) [ B ε ( x, u ( x + )) − B ε ( x, u ( x − ))]+ Z ] a,b [ φ ( x ) ( D x B ε )( x, u ( x )) dx =: D ε + J ε + I ε , and we study the convergence of each one of the three terms D ε , J ε , I ε appearing atthe right-hand side as ε → + .The limits of D ε and J ε are computed respectively in Steps 2 and 3 following thelines of [12].The limit of I ε is far more difficult to analyze, because of the possible interactionbetween the jump set of u and the jump set of B ( · , u ). In Step 4 we compute thislimit in the special case of u piecewise constant. Finally, the general case is provedin Step 5 relying on a carefully chosen approximation of a BV function by means ofpiecewise constant functions, whose construction has been shown in Lemma 3.2. Step 1. Fix φ ∈ C c (] a, b [) and let ϕ ε = ϕ ε ( x ) be a standard family of mollifiers.Let us define B ε ( x, w ) := Z ] a,b [ ϕ ε ( x − y ) B ( y, w ) dy for x ∈ ] a ′ , b ′ [ and w ∈ R d , where supp φ ⊂ [ a ′ , b ′ ] ⊂ ] a, b [ , and 0 < ε < min { b − b ′ , a − a ′ } .We claim that B ε ∈ C (] a ′ , b ′ [ × R d ). Firstly we prove that D x B ε is locally Lip-schitz continuous in ] a ′ , b ′ [ × R d . In fact, by hypothesis (A3) for every compact set D ⊆ ] a ′ , b ′ [ × R d and for every ( x , w ) , ( x , w ) ∈ D there exists a constant C D suchthat | ( D x B ε )( x , w ) − ( D x B ε )( x , w ) | = (cid:12)(cid:12)(cid:12) Z ] a,b [ (cid:2) ϕ ′ ε ( x − y ) B ( y, w ) − ϕ ′ ε ( x − y ) B ( y, w ) (cid:3) dy (cid:12)(cid:12)(cid:12) ≤ (cid:12)(cid:12)(cid:12) Z ] a,b [ (cid:2) ϕ ′ ε ( x − y ) − ϕ ′ ε ( x − y ) (cid:3) B ( y, w ) dy (cid:12)(cid:12)(cid:12) + (cid:12)(cid:12)(cid:12) Z ] a,b [ ϕ ′ ε ( x − y ) (cid:2) B ( y, w ) − B ( y, w ) (cid:3) dy (cid:12)(cid:12)(cid:12) ≤ C D ε (cid:0) | x − x | + | w − w | (cid:1) . Moreover, we prove that D w B ε is continuous in ] a ′ , b ′ [ × R d . In fact, for every sequence chain rule formula in BV x n , w n ) converging to ( x, w ) in ] a ′ , b ′ [ × R d we have | ( D w B ε )( x n , w n ) − ( D w B ε )( x, w ) | = (cid:12)(cid:12)(cid:12) Z ] a,b [ (cid:2) ϕ ε ( x n − y )( D w B )( y, w n ) − ϕ ε ( x − y )( D w B )( y, w ) (cid:3) dy (cid:12)(cid:12)(cid:12) ≤ Z ] a,b [ (cid:12)(cid:12) ϕ ε ( x n − y ) − ϕ ε ( x − y ) (cid:12)(cid:12) (cid:12)(cid:12) ( D w B )( y, w n ) (cid:12)(cid:12) dy + Z ] a,b [ (cid:12)(cid:12) ϕ ε ( x − y ) (cid:12)(cid:12) (cid:12)(cid:12) ( D w B )( y, w n ) − ( D w B )( y, w ) (cid:12)(cid:12) dy . The first integral tends to 0, as n → ∞ , since by (A3) Z ] a,b [ (cid:12)(cid:12) ϕ ε ( x n − y ) − ϕ ε ( x − y ) (cid:12)(cid:12) (cid:12)(cid:12) ( D w B )( y, w n ) (cid:12)(cid:12) dy ≤ Cε D M | x n − x | , and the second one tends to 0, as n → ∞ , by the continuity of the function ( D w B )( y, · )for a.e. y ∈ ] a, b [, the boundedness of D w B and by the Lebesgue dominated conver-gence theorem.Let u ∈ BV (] a, b [; R d ) and define v ε ( x ) := B ε ( x, u ( x )) , x ∈ ] a ′ , b ′ [ . Since B ε ∈ C (] a ′ , b ′ [ × R d ) we can apply the chain rule formula (see Theorem 3.96 in[3]) to the composition of the function B ε with the BV map x ( x, u ( x )), concludingthat v ε ∈ BV (] a ′ , b ′ [) and Z ] a ′ ,b ′ [ φ ′ ( x ) v ε ( x ) dx = − Z ] a ′ ,b ′ [ φ ( x ) ( D x B ε ) (cid:0) x, u ( x )) dx − Z ] a ′ ,b ′ [ φ ( x )( D u B ε )( x, u ( x )) · d e D u ( x ) − X x ∈ J u ∩ ] a ′ ,b ′ [ φ ( x ) (cid:0) B ε ( x, u ( x + )) − B ε ( x, u ( x − )) (cid:1) = − Z ] a ′ ,b ′ [ φ ( x ) ( D x B ε ) (cid:0) x, u ( x )) dx − d X i =1 Z ] a ′ ,b ′ [ φ ( x )( D w i B ε )( x, u ( x )) d e Du i ( x ) − d X i =1 X x ∈ J ui ∩ ] a ′ ,b ′ [ φ ( x ) (cid:0) u i ( x + ) − u i ( x − ) (cid:1) Z ( D w i B ε ) (cid:0) x, w s ( x )) (cid:1) ds , (4.7)where e D u and e Du i denote the diffuse parts of the measures D u and Du i respectively,and w s ( x ) := u ( x − ) + s ( u ( x + ) − u ( x − )).Since B is locally bounded and the functions B ε ( · , w ) converge a.e. in ] a ′ , b ′ [ to B ( · , w ), by Lebesgue dominated convergence theorem we getlim ε → + Z ] a,b [ φ ′ ( x ) B ε ( x, u ( x )) dx = Z ] a,b [ φ ′ ( x ) B ( x, u ( x )) dx . (4.8)2 G. Crasta and V. De Cicco Step 2. We shall prove the convergence of the diffuse part, i.e. for every i = 1 , . . . , d we prove thatlim ε → + Z ] a ′ ,b ′ [ φ ( x )( D w i B ε )( x, u ( x )) d e Du i = Z ] a,b [ φ ( x )( D w i B )( x, u ( x )) d e Du i . (4.9)Using the coarea formula (2.13), we get Z ] a ′ ,b ′ [ φ ( x )( D w i B ε )( x, u ( x ) , . . . , u i ( x ) , . . . , u d ( x )) d e Du i (4.10)= Z ] a ′ ,b ′ [ ∩ C ui φ ( x )( D w i B ε )( x, u ( x ) , . . . , u i ( x ) , . . . , u d ( x )) e Du i | Du i | ( x ) d | Du i | = Z + ∞−∞ dt Z { u i − ≤ t ≤ u i + }∩ C ui φ ( x )( D w i B ε )( x, u ( x )) e Du i | Du i | ( x ) d H = Z + ∞−∞ dt Z { u i = t }∩ C ui φ ( x )( D w i B ε )( x, u ( x ) , . . . , t, . . . , u d ( x )) e Du i | Du i | ( x ) d H . Now, by (A4) we have that for every i = 1 , . . . , d and for every w ∈ R d ( D w i B ε )( x, w ) → ( D w i B ) ∗ ( x, w ) ∀ x ∈ ] a, b [ (4.11)as ε → 0. Therefore, for a.e. t ∈ R , we havelim ε → Z { u i = t }∩ C ui φ ( x )( D w i B ε )( x, u , . . . , t, . . . , u d ) e Du i | Du i | d H = Z { u i = t }∩ C ui φ ( x )( D w i B ) ∗ ( x, u , . . . , t, . . . , u d ) e Du i | Du i | d H . From this equation, using the local boundedness of ( D w i B ) ∗ and the fact that, by thecoarea formula (2.13), Z + ∞−∞ H (cid:0) { u i = t } ∩ C u i (cid:1) dt = | Du i | ( C u i ) < ∞ , we can pass to the limit in (4.10) and by Lebesgue dominated convergence theoremwe get lim ε → Z ] a ′ ,b ′ [ φ ( x )( D w i B ε )( x, u ( x ) , . . . , u i ( x ) , . . . , u d ( x )) d e Du i = Z + ∞−∞ dt Z { u i = t }∩ C ui φ ( x )( D u i B ) ∗ ( x, u , . . . , t, . . . , u d ) d e Du i . From this equation, using the coarea formula (2.13) again, we immediately get (4.9). Step 3. We shall prove the convergence of the jump part, i.e. for every i = 1 , . . . , d we prove thatlim ε → + X x ∈ J ui ∩ ] a ′ ,b ′ [ φ ( x ) (cid:0) u i ( x + ) − u i ( x − ) (cid:1) Z ( D w i B ε ) (cid:0) x, w s ( x )) (cid:1) ds = X x ∈ J ui ∩ ] a ′ ,b ′ [ φ ( x ) (cid:0) u i ( x + ) − u i ( x − ) (cid:1) Z ( D w i B ) ∗ (cid:0) x, w s ( x )) (cid:1) ds , (4.12) chain rule formula in BV w s ( x ) := u ( x − ) + s ( u ( x + ) − u ( x − )). Let us fix i ∈ { , . . . , d } , and let J ′ u i := J u i ∩ ] a ′ , b ′ [= { y j } j ∈ N . For every h ∈ N there exists k ( h ) ∈ N such that ∞ X j = k ( h )+1 | u i ( y j + ) − u i ( y j − ) | < h . Then the following estimate holds: (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) X x ∈ J ′ ui φ ( x )( u i ( x + ) − u i ( x − )) Z (cid:16) ( D w i B ε )( x, w s ( x )) − ( D w i B ) ∗ ( x, w s ( x )) (cid:17) ds (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ≤ k φ k ∞ k ( h ) X j =1 (cid:12)(cid:12) u i ( y j + ) − u i ( y j − ) (cid:12)(cid:12)Z (cid:12)(cid:12)(cid:12) ( D w i B ε )( y j , w s ( y j )) − ( D w i B ) ∗ ( y j , w s ( y j )) (cid:12)(cid:12)(cid:12) ds + k φ k ∞ X j>k ( h ) (cid:12)(cid:12) u i ( y j + ) − u i ( y j − ) (cid:12)(cid:12)Z (cid:12)(cid:12)(cid:12) ( D w i B ε )( y j , w s ( y j )) − ( D w i B ) ∗ ( y j , w s ( y j )) (cid:12)(cid:12)(cid:12) ds ≤ k φ k ∞ Z k ( h ) X j =1 (cid:12)(cid:12) u i ( y j + ) − u i ( y j − ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12) ( D w i B ε )( y j , w s ( y j )) − ( D w i B ) ∗ ( y j , w s ( y j )) (cid:12)(cid:12)(cid:12) ds + 2 C k φ k ∞ X j>k ( h ) (cid:12)(cid:12) u i ( y j + ) − u i ( y j − ) (cid:12)(cid:12) , where C := k D w i B k L ∞ (] a ′ ,b ′ [ × ] − M,M [) and k u i k ∞ ≤ M . By Lebesgue dominatedconvergence theorem, the first integral is infinitesimal as ε → 0, since ( D w i B ε )( x, w )and ( D w i B ) ∗ ( x, w ) are locally bounded functions and for every x ∈ ] a ′ , b ′ [ and w ∈ R d we have that ( D w i B ε )( x, w ) → ( D w i B ) ∗ ( x, w ), as ε → 0. Therefore, letting first ε tend to zero and then h tend to ∞ , we immediately obtain (4.12). Step 4. In this step, we consider a piecewise constant function u : ] a, b [ → R d of theform u ( x ) = N X i =0 v i χ ∗ [ a i ,a i +1 ] ( x ) , where v , . . . , v N ∈ R d , a = a < a < . . . < a N < a N +1 = b and we prove thatlim ε → Z ] a ′ ,b ′ [ φ ( x ) ( D x B ε )( x, u ( x )) dx = Z ] a ′ ,b ′ [ φ ( x ) ( ∇ x B )( x, u ( x )) dx + Z ] a ′ ,b ′ [ φ ( x ) d ( D cx B )( · , u ) d λ d λ + X x ∈N ∪ J u φ ( x ) (cid:20) B ( x + , u ( x + )) + B ( x + , u ( x − ))2 − B ( x − , u ( x + ) + B ( x − , u ( x − ))2 (cid:21) . (4.13)In order to simplify the notation, let us denote by χ i the characteristic function4 G. Crasta and V. De Cicco χ [ a i ,a i +1 ] . From the very definition of u and (2.14), we have that I ε := Z ] a ′ ,b ′ [ φ ( x ) ( D x B ε )( x, u ( x )) dx = N X i =0 Z ] a ′ ,b ′ [ φ ( x ) χ ∗ i ( x ) ( D x B ε )( x, v i ) dx = − N X i =0 Z ] a ′ ,b ′ [ B ε ( x, v i ) dD ( φχ i )= − N X i =0 Z ] a ′ ,b ′ [ φ ′ ( x ) χ i ( x ) B ε ( x, v i ) dx + N X i =0 (cid:2) φ ( a i +1 ) B ε ( a i +1 , v i ) − φ ( a i ) B ε ( a i , v i ) (cid:3) . Passing to the limit as ε → ε → I ε = − N X i =0 Z ] a,b [ φ ′ ( x ) χ i ( x ) B ( x, v i ) dx + N X i =0 (cid:2) φ ( a i +1 ) B ∗ ( a i +1 , v i ) − φ ( a i ) B ∗ ( a i , v i ) (cid:3) . (4.14)Let us consider the integrals at the right-hand side of (4.14). Using again (2.14) wehave that − Z ] a,b [ φ ′ ( x ) χ i ( x ) B ( x, v i ) dx = Z ] a,b [ φ ( x ) dD ( χ i B ( · , v i ))= Z ] a,b [ φ ( x ) χ ∗ i ( x ) d ( D x B )( · , v i ) + Z ] a,b [ φ ( x ) B ∗ ( x, v i ) dDχ i = Z ] a,b [ φ ( x ) χ ∗ i ( x ) d ( D x B )( · , v i ) + φ ( a i ) B ∗ ( a i , v i ) − φ ( a i +1 ) B ∗ ( a i +1 , v i ) . Substituting this expression into (4.14) we thus obtain I := lim ε → I ε = N X i =0 Z ] a,b [ φ ( x ) χ ∗ i ( x ) d ( D x B )( · , v i ) . Finally, let us decompose each measure ( D x B )( · , v i ) in the canonical way (4.1). It isnot difficult to check that I = N X i =0 Z ] a,b [ φ ( x ) χ ∗ i ( x ) ∇ x B ( x, v i ) dx + N X i =0 Z ] a,b [ φ ( x ) χ ∗ i ( x ) dD cx B ( · , v i ) d λ d λ + N X i =1 φ ( a i ) (cid:20) B ( a i + , v i ) + B ( a i + , v i − )2 − B ( a i − , v i ) + B ( a i − , v i − )2 (cid:21) + X x ∈N \ J u φ ( x ) [ B ( x + , u ( x )) − B ( x − , u ( x ))] . (4.15) chain rule formula in BV Z ] a,b [ φ ( x ) ∇ x B ( x, u ( x )) dx , Z ] a,b [ φ ( x ) ψ ( x, u ( x )) dλ . The last two summations take into account the jump points x ∈ J u and x ∈ N \ J u ,respectively. Again, it is not difficult to check that, in both cases, the correspondingterm can always be written as φ ( x ) (cid:20) B ( x + , u ( x + )) + B ( x + , u ( x − ))2 − B ( x − , u ( x + ) + B ( x − , u ( x − ))2 (cid:21) , so that (4.13) follows. Step 5. In this step, we shall prove that formula (4.13) holds for every function u ∈ BV (] a, b [; R d ), i.e. we prove that I = lim ε → + I ε , (4.16)where I ε := Z ] a ′ ,b ′ [ φ ( x ) ( D x B ε ) (cid:0) x, u ( x )) dx (4.17)and I := Z ] a,b [ φ ( x )( ∇ x B )( x, u ( x )) dx + Z ] a,b [ φ ( x ) ψ ( x, u ( x )) dλ + X x ∈N ∪ J u φ ( x ) (cid:20) B ( x + , u ( x + )) + B ( x + , u ( x − ))2 − B ( x − , u ( x + ) + B ( x − , u ( x − ))2 (cid:21) . Let u ∈ BV (] a, b [; R d ) and let ( u n ) n be the sequence of approximating piecewiseconstant functions given by Lemma 3.2 with P = N \ J u .Fixed ε > 0, we set I nε := Z ] a ′ ,b ′ [ φ ( x ) ( D x B ε ) (cid:0) x, u n ( x )) dx . (4.18)By Lebesgue dominated convergence theorem and the continuity of ( D x B ε ) (cid:0) x, · )(which follows by B ε ∈ C (] a ′ , b ′ [ × R d )), for every ε > I ε = lim n →∞ I nε . (4.19)More precisely, we claim that | I nε − I ε | ≤ k u n − u k ∞ µ M (] a, b [) k φ k ∞ ∀ ε > . (4.20)Namely, by hypothesis (A2) we have | I nε − I ε | ≤ Z ] a,b [ | φ ( x ) | h Z ] a,b [ ϕ ε ( x − y ) d (cid:12)(cid:12) ( D x B )( · , u n ( x )) − ( D x B )( · , u ( x )) (cid:12)(cid:12) ( y ) i dx ≤ Z ] a,b [ | φ ( x ) || u n ( x ) − u ( x ) | h Z ] a,b [ ϕ ε ( x − y ) dµ M ( y ) i dx ≤k u n − u k ∞ k φ k ∞ Z ] a,b [ h Z ] a,b [ ϕ ε ( x − y ) dx i dµ M ( y )= k u n − u k ∞ k φ k ∞ µ M (] a, b [) . (4.21)6 G. Crasta and V. De Cicco On the other hand, by Step 4 we have for every n ∈ N lim ε → + I nε = I n (4.22)where I n := Z ] a,b [ φ ( x )( ∇ x B )( x, u n ( x )) dx + Z ] a,b [ φ ( x ) ψ ( x, u n ( x )) dλ + S n , (4.23)and S n := X x ∈N ∪ J u n φ ( x ) h B ( x + , u n ( x + )) + B ( x + , u n ( x − ))2 + − B ( x − , u n ( x + ) + B ( x − , u n ( x − ))2 i . We claim that I = lim n →∞ I n . (4.24)By Remark 4.2 we have thatlim n →∞ Z ] a,b [ φ ( x )( ∇ x B )( x, u n ( x )) dx = Z ] a,b [ φ ( x )( ∇ x B )( x, u ( x )) dx and lim n →∞ Z ] a,b [ φ ( x ) ψ ( x, u n ( x )) dλ = Z ] a,b [ φ ( x ) ψ ( x, u ( x )) dλ . It remains to show that S n converges, as n → + ∞ , to S := X x ∈N ∪ J u φ ( x ) (cid:20) B ( x + , u ( x + )) + B ( x + , u ( x − ))2 − B ( x − , u ( x + ) + B ( x − , u ( x − ))2 (cid:21) . Let I = S n J u n , I = N \ I and I = I ∪ I . We recall that P = N \ J u and J u ⊂ I .Since, by construction, J u n ∩ P = ∅ for every n ∈ N , we have that I ∩ P = ∅ and N ∪ J u ⊂ I . Hence both summations in S n and S can be extended to the bigger set I = { x i } , since it is easy to check that the added terms are all zero. Thus we canwrite S n = X i ∈ N a ni , S = X i ∈ N a i , where a ni := φ ( x i ) (cid:20) B ( x i + , u n ( x i + )) + B ( x i + , u n ( x i − ))2 − B ( x i − , u n ( x i + ) + B ( x i − , u n ( x i − ))2 (cid:21) ,a i := φ ( x i ) (cid:20) B ( x i + , u ( x i + )) + B ( x i + , u ( x i − ))2 − B ( x i − , u ( x i + ) + B ( x i − , u ( x i − ))2 (cid:21) . Let R ≥ max {k u n k ∞ , k u k ∞ } and let M = B R (0). From assumption (A2) we havethat | B ( x + , w ) − B ( x − , w ) | ≤ | B ( x + , − B ( x − , | + Rµ M ( { x } ) , x ∈ ] a, b [ , w ∈ M . chain rule formula in BV | a ni | ≤ b i := k φ k ∞ (cid:2) | B ( x i + , − B ( x i − , | + Rµ M ( { x i } ) (cid:3) , ∀ n ∈ N , and X i ∈ N b i ≤ k φ k ∞ [ T V ( B ( · , Rµ M (] a, b [)] , ∀ n ∈ N , in order to prove that lim n S n = S , by Dominated convergence theorem, it is enoughto prove that lim n a ni = a i for every i ∈ N .We have three cases. If x i ∈ P = N \ J u , then u and every u n are continuous at x i . Moreover, for every n large enough, u n ( x i ) = u ( x i ), hence a ni = a i . If x i ∈ J u ,then for every n large enough we have that u n ( x i +) = u ( x i +), u n ( x i − ) = u ( x i − ),hence again a ni = a i . Finally, let us consider the case x i ∈ I \ J u . Since x i 6∈ N , thefunction B ( x i , · ) is continuous in R d . Moreover, since x i J u , also u is continuous at x i and u n ( x i +) , u n ( x i − ) → u ( x i ), so that lim n a ni = a i . Therefore (4.24) is proved.In order to prove (4.16), let us fix η > 0. By (4.20) and by (4.24) there exists n ∈ N such that | I n ε − I ε | < η ∀ ε > | I n − I | < η . Moreover by (4.22) there exists ε > | I n ε − I n | < η ∀ < ε < ε . Then | I ε − I | ≤ | I ε − I n ε | + | I n ε − I n | + | I n − I | < η ∀ < ε < ε . Therefore (4.16) is proved and this concludes Step 5.Finally, the thesis of the theorem is obtained by collecting all the Steps.In view to the applications to conservation laws (see Proposition 6.1) we needto generalize formula (4.2) in order to integrate a BV function with respect to themeasure ( B ( x, u ( x ))) x . Corollary 4.4. Let B :] a, b [ × R → R be a function satisfying the same assump-tions of Theorem 4.1. Let g :] a, b [ → R be a BV -function such that J g ⊆ N .Then for every u ∈ BV (] a, b [; R d ) and φ ∈ C (] a, b [) we have Z ] a,b [ φ ( x ) g ∗ ( x ) d ( B ( x, u ( x ))) x = Z ] a,b [ φ ( x ) g ( x )( ∇ x B )( x, u ( x )) dx + Z ] a,b [ φ ( x ) g ( x ) ψ ( x, u ( x )) dλ + Z ] a,b [ φ ( x ) g ( x )( D w B )( x, u ( x )) · ∇ u ( x ) dx + Z ] a,b [ φ ( x ) g ( x )( D w B )( x, u ( x )) · dD c u ( x )+ X x ∈N ∪ J u φ ( x ) g ∗ ( x ) [ B ( x + , u ( x + )) − B ( x − , u ( x − ))] . (4.25)8 G. Crasta and V. De Cicco Proof . Let g ε = g ∗ ϕ ε be the standard mollified functions of the BV -function g .We recall that g ε pointwise converges (everywhere) in ] a, b [ to the precise representa-tive g ∗ , as ε → φ ( x ) g ε ( x ) as test function. Theconclusion follows by Lebesgue dominated convergence Theorem.In the next corollaries we consider B ( x, w ) with a particular structure. Corollary 4.5. Let K ∈ BV (] a, b [) and f ∈ C ( R d ) . Then for every u ∈ BV (] a, b [; R d ) the function v :] a, b [ → R , defined by v ( x ) := K ( x ) f ( u ( x )) , x ∈ ] a, b [ , belongs to BV (] a, b [) , and for any φ ∈ C (] a, b [) we have Z ] a,b [ φ ′ ( x ) v ( x ) dx = − Z ] a,b [ φ ( x )( f ( u )) ∗ ( x ) dDK ( x ) − Z ] a,b [ φ ( x ) K ( x )( ∇ f )( u ( x )) · ∇ u ( x ) dx − Z ] a,b [ φ ( x ) K ( x )( ∇ f )( u ( x )) · dD c u ( x ) − X x ∈ J u φ ( x ) K ∗ ( x ) [ f ( u ( x + )) − f ( u ( x − ))] , (4.26) where ( f ( u )) ∗ and K ∗ are the precise representatives of the BV functions f ( u ) and K respectively.Proof . It is sufficient to observe that the function B ( x, w ) := K ( x ) f ( w ) satis-fies all the assumptions of Theorem 4.1 . For instance, hypothesis (A2) is satisfiedsince for every compact set M ⊂ R d we can choose µ M := C M | DK | , with C M =max w ∈ M |∇ f ( w ) | , and hypothesis (A5) is satisfied since we can choose λ := | DK | . Corollary 4.6. Let f : R × R d → R and K :] a, b [ → R be two functions satisfying(i) K ∈ BV (] a, b [) ;(ii) the function f = f ( y, w ) belongs to C ( R × R d ) ;(iii) the function x f w ( K ( x ) , w ) belongs to BV (] a, b [) for every w ∈ R d ;(iv) for every compact set D ⊂ R × R d there exists a constant L D such that | f y ( y, w ) − f y ( y, w ′ ) | ≤ L D | w − w ′ | ∀ ( y, w ) , ( y, w ′ ) ∈ D and | f w ( y, w ) − f w ( y ′ , w ) | ≤ L D | y − y ′ | ∀ ( y, w ) , ( y ′ , w ) ∈ D . Then for every u ∈ BV (] a, b [; R d ) the function v :] a, b [ → R , defined by v ( x ) := f ( K ( x ) , u ( x )) , x ∈ ] a, b [ , belongs to BV (] a, b [) , and for any φ ∈ C (] a, b [) we have Z ] a,b [ φ ′ ( x ) v ( x ) dx = − Z ] a,b [ φ ( x ) f y ( K ( x ) , u ( x )) · d e DK ( x ) − Z ] a,b [ φ ( x ) f w ( K ( x ) , u ( x )) · d e D u ( x ) − X x ∈ J u ∪ J K φ ( x ) [ f ( K ( x + ) , u ( x + )) − f ( K ( x − ) , u ( x − ))] . (4.27) chain rule formula in BV Proof . We observe that the function B ( x, w ) = f ( K ( x ) , w ) satisfies all the as-sumptions of Theorem 4.1 . In particular, N = J K , D cx B ( · , w ) = f y ( K ( x ) , w ) D c K ≪ | D c K | = λ and ψ ( x, w ) = f y ( K ( x ) , w ) D c K | D c K | ( x ). 5. Comparison with other chain rule formulas. In [12] it was proved achain rule formula for function u : R N → R . In Theorem 5.1 we recall this formulawhich coincides to formula (4.2) in the case of d = 1 = N . Although the two formulaslook like very different, we explicitely show that they concide for piecewise constantfunctions. Theorem 5.1. Let B :] a, b [ × R → R be a locally bounded Borel function. Assumethat B ( x, 0) = 0 for all x ∈ ] a, b [ and(i) for all t ∈ R the function B ( · , t ) ∈ BV (] a, b [) ; (ii) for all x ∈ ] a, b [ the function B ( x, · ) belongs to C ( R ) ; (iii) the function D t B is locally bounded, for all t ∈ R the function ( D t B ) ( · , t ) belongs to BV (] a, b [) and for every compact set M ⊂ R Z M | D x ( D t B )( · , t ) | (] a, b [) dt < + ∞ . Then, for every u ∈ BV (] a, b [) the composite function v ( x ) := B ( x, u ( x )) , x ∈ ] a, b [ ,belongs to BV loc (] a, b [) and for any φ ∈ C (] a, b [) we have Z ] a,b [ φ ′ ( x ) v ( x ) dx = − Z + ∞−∞ dt Z ] a,b [ sgn( t ) χ ∗ Ω u,t ( x ) φ ( x ) dD x ( D t B )( · , t ) (5.1) − Z ] a,b [ φ ( x )( D t B )( x, u ( x )) ∇ u ( x ) dx − Z ] a,b [ φ ( x )( D t B )( x, u ( x )) dD c u ( x ) − X x ∈ J u φ ( x ) [ B ∗ ( x, u ( x + )) − B ∗ ( x, u ( x − ))] , where Ω u,t = { x ∈ ] a, b [: t belongs to the segment of endpoints and u ( x ) } and χ ∗ Ω u,t and B ∗ ( · , t ) are, respectively, the precise representatives of the BV functions χ Ω u,t and B ( · , t ) .Proof . It is a consequence of Theorem 1.1 in [12], with N = 1 and B ( x, t ) = Z t b ( x, s ) ds . We recall that in our case for the approximate limits u + ( x ) and u − ( x ) we have u + ( x ) = ν u ( x ) u ( x + ) and u − ( x ) = ν u ( x ) u ( x − ), where ν u = ± G. Crasta and V. De Cicco Remark 5.2. When d = 1 , if we assume also all the hypotheses of Theorem 4.1and we compare formulas (5.1) and (4.2), we conclude that Z + ∞−∞ dt Z ] a,b [ sgn( t ) χ ∗ Ω u,t ( x ) φ ( x ) dD x ( D t B )( · , t ) = Z ] a,b [ φ ( x )( ∇ x B )( x, u ( x )) dx + Z ] a,b [ φ ( x ) ψ ( x, u ( x )) dλ + X x ∈N φ ( x ) (cid:20) B ( x + , u ( x + )) + B ( x + , u ( x − ))2 − B ( x − , u ( x + )) + B ( x − , u ( x − ))2 (cid:21) . (5.2) For piecewise constant functions this formula can be proved by using formula (4.15)and the following proposition. Proposition 5.3. For every piecewise constant function u : ] a, b [ → R of the form u ( x ) = N X i =0 v i χ ∗ [ a i ,a i +1 ] ( x ) , where v , . . . , v N ∈ R , a = a < a < . . . < a N < a N +1 = b , we have that Z + ∞−∞ dt Z ] a,b [ sgn( t ) χ ∗ Ω u,t ( x ) φ ( x ) dD x ( D t B )( · , t ) = N X i =0 Z ] a,b [ φ ( x ) χ ∗ [ a i ,a i +1 ] ( x ) d ( D x B )( · , v i ) . Proof . It is not restrictive to assume that u ≥ . By the Leibnitz formula (2.14)we have that Z + ∞−∞ dt Z ] a,b [ sgn( t ) χ ∗ Ω u,t ( x ) φ ( x ) dD x ( D t B )( · , t )= Z + ∞ dt Z ] a,b [ φ ( x ) dD x ( χ Ω u,t ( D t B ))( · , t ) − Z + ∞ dt Z ] a,b [ φ ( x )( D t B ) ∗ ( x, t ) dDχ Ω u,t ( x )=: I + I . Since B ( · , 0) = 0, we have that I = − Z + ∞ dt Z ] a,b [ φ ′ ( x ) χ Ω u,t ( x ) ( D t B )( x, t ) dx = − Z ] a,b [ φ ′ ( x ) Z u ( x )0 ( D t B )( x, t ) dt ! dx = − Z ] a,b [ φ ′ ( x ) B ( x, u ( x )) dx = − N X i =0 Z ] a,b [ φ ′ ( x ) χ ] a i ,a i +1 [ ( x ) B ( x, v i ) dx = N X i =0 "Z ] a,b [ φ ( x ) χ ∗ [ a i ,a i +1 ] ( x ) d ( D x B )( x, v i ) − φ ( a i +1 ) B ∗ ( a i +1 , v i ) + φ ( a i ) B ∗ ( a i , v i ) . (5.3) chain rule formula in BV I , let us observe that Dχ Ω u,t is an atomic measurewith support contained in { a , . . . , a N } . Moreover Dχ Ω u,t ( { a i } ) = , if v i − < v i and t ∈ [ v i − , v i [ , − , if v i − > v i and t ∈ [ v i , v i − [ , , otherwise . Therefore, by Fubini’s theorem we obtain I = − N X i =1 φ ( a i ) Z v i v i − ( D t B ) ∗ ( a i , t ) dt . We claim that Z v i v i − ( D t B ) ∗ ( a i , t ) dt = B ∗ ( a i , v i ) − B ∗ ( a i , v i − ) , so that I = − N X i =1 φ ( a i )[ B ∗ ( a i , v i ) − B ∗ ( a i , v i − )] . (5.4)Namely, since B ( · , t ) ∈ BV (] a, b [), ( D t B ) ( · , t ) ∈ BV (] a, b [) and | ( D t B )( x, t ) | ≤ D M for every x ∈ ] a, b [, we have that Z v i v i − ( D t B ) ∗ ( a i , t ) dt = 12 Z v i v i − (cid:20) lim x → a i + ( D t B )( x, t ) + lim x → a i − ( D t B )( x, t ) (cid:21) dt = lim x → a i + Z v i v i − ( D t B )( x, t ) dt + lim x → a i − Z v i v i − ( D t B )( x, t ) dt = lim x → a i + 12 [ B ( x, v i ) − B ( x, v i − )] + lim x → a i − 12 [ B ( x, v i ) − B ( x, v i − )]= B ∗ ( a i , v i ) − B ∗ ( a i , v i − ) . Finally, the conclusion follows from (5.3) and (5.4). 6. An application to conservation laws. In this section we shall apply thechain rule formula in order to study a scalar conservation law where the flux dependsdiscontinuously on the space variable: u t ( x, t ) + B ( x, u ( x, t )) x = 0 , ( x, t ) ∈ R × [0 , + ∞ ) , (6.1)where B : R × R → R is a function satisfying the assumptions of Theorem 4.1 (with] a, b [= R ).For every x ∈ R we define the set of pairs ( u − , u + ) satisfying the Rankine-Hugoniot condition A x = { ( u − , u + ) ∈ R × R : B ( x − , u − ) = B ( x + , u + ) } . We define an entropy-flux pair ( η, q ) associated to (6.1), as a pair of functions η, q : R × R → R such that:2 G. Crasta and V. De Cicco (E1) for every x ∈ R the function η ( x, · ) is convex and η u is locally bounded in R × R ; moreover, for every u ∈ R the functions η ( · , u ), η u ( · , u ) belong to BV ( R ) and their jump set is contained in J K ;(E2) q ( x, · ) ∈ Lip loc ( R ) for every x ∈ R , and q ( · , u ) ∈ BV ( R ), J q ( · ,u ) ⊆ J K forevery u ∈ R ;(E3) η u ( x, u ) B u ( x, u ) = q u ( x, u ) for every x ∈ R \ J K and u ∈ R ;(E4) q ( x + , u + ) − q ( x − , u − ) ≤ x ∈ R and every ( u − , u + ) ∈ A x . Proposition 6.1. Let u be a bounded piecewise C solution of (6.1) with J u ( · ,t ) ⊆ J K for every t ∈ ]0 , T [ , and let ( η, q ) be an entropy-entropy flux pair associated to (6.1) .Then u satisfies the following inequality ( η ( x, u )) t + ( q ( x, u )) x ≤ in the sense of measures, i.e. ZZ R × ]0 ,T [ φ ( x, t ) d [( η ( x, u ( x, t ))) t + ( q ( x, u ( x, t ))) x ] ≤ for every function φ : R × ]0 , T [ → [0 , ∞ [ continuous with compact support.Proof . We remark that, by the chain rule formula and since η does not dependon t , we have ( η ( x, u ( x, t ))) t = η ∗ u ( x, u ) u t ( x, t ) dt in the sense of measure. Then Z Z R × ]0 ,T [ φ ( x, t ) d ( η ( x, u )) t = Z Z R × ]0 ,T [ φ ( x, t ) η ∗ u ( x, u ) u t ( x, t ) dt , so that Z Z R × ]0 ,T [ φ ( x, t ) d ( η ( x, u )) t + ( q ( x, u )) x = Z Z R × ]0 ,T [ φ ( x, t ) ( η ∗ u ( x, u )) u t ( x, t ) dt + Z Z R × ]0 ,T [ φ ( x, t ) d ( q ( x, u )) x , where ( η u ( x, u )) ∗ is the precise representative of the composition of η u ( x, · ) with thefunction u . By (6.1) we have that u t ( x, t ) dt = − ( B ( x, u )) x in the sense of measures,i.e. Z Z R × ]0 ,T [ φ ( x, t ) u t ( x, t ) dt = − Z Z R × ]0 ,T [ φ ( x, t ) d ( B ( x, u )) x . Since the jumps of η u ( · , u )) are contained in J K , reasoning as in the proof of Corol-lary 4.4, we have that ( η u ( x, u )) ∗ u t ( x, t ) dt = − ( η u ( x, u )) ∗ ( B ( x, u )) x in the sense ofmeasures, i.e. Z Z R × ]0 ,T [ φ ( x, t ) ( η u ( x, u )) ∗ u t ( x, t ) dt = − Z Z R × ]0 ,T [ φ ( x, t ) ( η u ( x, u )) ∗ d ( B ( x, u )) x . Hence Z Z R × ]0 ,T [ φ ( x, t ) d [( η ( x, u )) t + ( q ( x, u )) x ]= − Z Z R × ]0 ,T [ φ ( x, t ) ( η u ( x, u )) ∗ d ( B ( x, u )) x + Z Z R × ]0 ,T [ φ ( x, t ) d ( q ( x, u )) x , chain rule formula in BV η u ( x, u )) ∗ B ( x, u ) x ≥ ( q ( x, u )) x in the sense ofmeasures, i.e. for every nonnegative function φ ∈ C c ( R × ]0 , T [) Z T (cid:20)Z R φ ( x, t )( η u ( x, u )) ∗ d B ( x, u ) x (cid:21) dt ≥ Z T (cid:20)Z R φ ( x, t ) d q ( x, u ) x (cid:21) dt . We use the chain rule formula (see Corollary 4.4) and condition (E3) to obtain I := Z T (cid:20)Z R φ ( x, t )( η u ( x, u )) ∗ d B ( x, u ) x (cid:21) dt = Z T (cid:20)Z R φ ( x, t ) η u ( x, u ) B u ( x, u ( x )) · d e Du ( x ) (cid:21) dt + Z T " X x ∈ J K φ ( x, t )( η u ( x, u )) ∗ (cid:16) B ( x + , u ( x + )) − B ( x − , u ( x − )) (cid:17) dt . We remark that the last term vanishes by the Rankine-Hugoniot condition. Using(E4) we obtain I ≥ Z T (cid:20)Z R φ ( x, t ) q u ( x, u ( x )) · d e Du ( x ) (cid:21) dt + Z T h X x ∈ J u φ ( x, t ) (cid:16) q ( x + , u ( x + )) − q ( x − , u ( x − )) (cid:17)i dt = Z T (cid:20)Z R φ ( x, t ) d q ( x, u ) x (cid:21) dt . This concludes the proof.We consider the partially adapted Kruzkov entropies introduced by Audusse andPerthame for discontinuous flux (see formula (1.3) in [4]).In addition to the assumptions on the function B stated in Theorem 4.1, we alsoassume thatfor every x ∈ R , the map B ( x, · ) is a one to one function from R to R . (6.4)Given α ∈ R , by assumption (6.4) there exists a unique function c α : D α → R ,defined on a (possibly empty) set D α ⊂ R , such that B ( x, c α ( x )) = α for every x ∈ D α . Proposition 6.2. For every α ∈ R such that c α is defined in R , let us definethe adapted Kruzkov entropy η ( α ) ( x, u ) := | u − c α ( x ) | and the corresponding flux q ( α ) ( x, u ) := (cid:0) B ( x, u ) − α (cid:1) (sgn( u − c α ( x ))) ∗ . We assume that for every x ∈ J K and for every ( u − , u + ) ∈ A x we have ( sgn ( u − − c α ( x − ))) ∗ = ( sgn ( u + − c α ( x + ))) ∗ . (6.5) Then we have G. Crasta and V. De Cicco • (a) ( η ( α ) , q ( α ) ) is an entropy-flux pair; in particular, the entropy inequality ∂ t | u − c α ( x ) | + ∂ x [ (cid:0) B ( x, u ) − α (cid:1) ( sgn ( u − c α ( x ))) ∗ ] ≤ holds in the sense of distributions; • (b) (6.2) holds for every entropy-flux pair ( η, q ) if and only if (6.6) holds forevery α as above.Proof . For every x ∈ R and u = c α ( x ) one has q ( α ) u ( x, u ) = (sgn( u − c α ( x )) ∗ B u ( x, u ).Then, since ( η ( α ) u ( x, u )) ∗ = (sgn( u − c α ( x ))) ∗ , we obtain η ( α ) u ( x, u ) B u ( x, u ) = q ( α ) u ( x, u ).Moreover, for every x ∈ R and every ( u − , u + ) ∈ A x satisfying (6.5) we have that q ( α ) ( x + , u + ) − q ( α ) ( x − , u − ) = (sgn( u + − c α ( x ))) ∗ (cid:0) B ( x + , u + ) − α (cid:1) − (sgn( u − − c α ( x ))) ∗ (cid:0) B ( x − , u − ) − α (cid:1) = 0 . In order to prove (b), let u be a bounded BV solution to (6.1). If u satisfies (6.2)for every entropy-flux pair ( η, q ), then from (a) it satisfies also (6.6) for every α .Conversely, assume now that u satisfies also (6.6) for every α . Let ( η, q ) be anentropy-flux pair, and let φ ( x, t ) be a non-negative test function. We have to provethat (6.3) holds.Assume that | u ( x, t ) | ≤ M for every ( x, t ), supp φ ⊂ ] a, b [ × ]0 , T [, and | B ( x, u ) | ≤ C for every ( x, u ) ∈ ] a, b [ × ] − M, M [. Let us fix a positive integer number N , and forevery x ∈ [ a, b ] define I ( x ) := { i ∈ Z : | i | ≤ N, α Ni := iC/N ∈ Range B ( x, · ) } ,m ( x ) := min I ( x ) , n ( x ) := max I ( x ) ,c Ni ( x ) := c α Ni ( x ) , i = m ( x ) , . . . , n ( x ) . We are going to approximate η (and so q ) by an entropy η N of the form η N ( x, u ) := a N ( x ) + b N ( x ) u + n ( x ) − X i = m ( x )+1 b Ni ( x ) | u − c Ni ( x ) | = a N ( x ) + b N ( x ) u + n ( x ) − X i = m ( x )+1 b Ni ( x ) η ( α Ni ) ( x, u ) , (6.7)where b Ni ( x ) ≥ i and x . Indeed, if we define δ Ni ( x ) := η ( x, c Ni +1 ( x )) − η ( x, c Ni ( x )) c Ni +1 ( x ) − c Ni ( x ) , x ∈ R , i = m ( x ) , . . . , n ( x ) − , and b N ( x ) := δ Nm ( x ) ( x ) + δ Nn ( x ) − ( x )2 ,b Ni ( x ) := δ Ni ( x ) − δ Ni − ( x )2 , i = m ( x ) + 1 , . . . , n ( x ) − ,a N ( x ) := η ( x, c Nm ( x ) ( x )) − b N ( x ) c Nm ( x ) ( x ) − n ( x ) − X i = m ( x )+1 b Ni ( x )[ c Ni ( x ) − c Nm ( x ) ( x )] , chain rule formula in BV b Ni ( x ) ≥ η N ( x, · ) is a convex piecewise affine function coinciding with η ( x, · ) in the points u i = c Ni ( x ), i = m ( x ) , . . . , n ( x ).The flux associated to η N is the function q N ( x, u ) := b N ( x ) B ( x, u ) + n ( x ) − X i = m ( x )+1 b Ni ( x ) q ( α Ni ) ( x, u ) . (6.8)For every N ∈ N we have that Z Z R × [0 ,T ] φ ( x, t )[( η N ( x, u ( x.t ))) t + ( q N ( x, u ( x, t ))) x ] dx dt = Z Z R × [0 ,T ] φ ( x, t ) b N ( x )[ u t ( x, t ) + ( B ( x, u ( x, t ))) x ] dx dt + n ( x ) − X i = m ( x )+1 Z Z R × [0 ,T ] φ ( x, t ) b Ni ( x )[( η ( α Ni ) ( x, u ( x, t ))) t + ( q ( α Ni ) ( x, u ( x, t ))) x ] dx dt , (6.9)where we recall that η ( α Ni ) ( x, u ) = | u − c α Ni ( x ) | and q ( α Ni ) ( x, u ) := (cid:0) B ( x, u ) − c α Ni ( x ) (cid:1) (sgn( u − c α Ni ( x )) ∗ . We recall that, given a non-negative measure µ (i.e. µ ( φ ) ≥ φ ≥ µ b ( φ ) := Z Z R × [0 ,T ] φ ( x, t ) b ( x, t ) dµ ( x, t ) , where b is a non-negative Borel function, then µ b is also a non-negative measure (see[15, Ch. 7]) . Hence from (6.1), (6.6) and (6.9) and the fact that the functions b Ni arenon-negative, we have Z Z R × [0 ,T ] φ ( x, t )[( η N ( x, u ( x.t ))) t + ( q N ( x, u ( x, t ))) x ] dx dt ≤ . 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