A relaxation result in BV for integral functionals with discontinuous integrands

ESAIM: Control, Optimisation and Calculus of Variations

Vol. 13, N^o 2, 2007, pp. 396–412 www.edpsciences.org/cocv

DOI: 10.1051/cocv:2007015

A RELAXATION RESULT IN BV FOR INTEGRAL FUNCTIONALS WITH DISCONTINUOUS INTEGRANDS

Micol Amar

, Virginia De Cicco

and Nicola Fusco

Abstract. We prove a relaxation theorem in BV for a non coercive functional with linear growth. No continuity of the integrand with respect to the spatial variable is assumed.

Mathematics Subject Classification. 49J45, 26B30.

Received November 11, 2005. Revised March 28, 2006.

1. Introduction

In this paper we study the relaxation in BV(Ω) of an integral functional of the type F(u) =

Ωf(x, u(x),∇u(x)) dx, whereuis a scalar function fromW^1,1(Ω).

In recent years there has been a renewed interest in the L¹-lower semicontinuity of such an integral functional and of its BV counterpart

F(u) =

Ωf(x, u,∇u)dx+

Ωf^∞

x,u, D^cu

|D^cu|

d|D^cu|+

Ju∩Ω

u⁺(x)

u⁻(x) f^∞(x, s, νu) ds

dH^N−1(x)

with the aim of weakening the regularity assumptions on the integrandf with respect to the spatial variablex(see [7–9, 18, 20, 21]). Roughly speaking, one can show that the L¹-lower semicontinuity still holds if one replaces the classical continuity and coerciveness assumptions with the weak differentiability off with respect tox. Therefore the results proved in the above papers suggest that a similar assumption should be also enough to prove that the relaxation in BV of the functionalF is represented by F.

In this paper we prove that this representation formula actually holds (see Th. 6.1) under the assumption that for all (s, ξ) ∈ IR×IR^N the functionf(·, s, ξ) is weakly differentiable and coincides H^N−1-a.e. with its precise representative, i.e., it is H^N−1-a.e. approximately continuous in Ω. Notice that though the functional F does not change its values if we modify f in a subset of zero Lebesgue measure of Ω, this modification may affect the values ofF on BV(Ω). Therefore, if f is not assumed to beH^N−1-a.e. approximately continuous with respect to x, then it is not true in general that the relaxation ofF is represented by F. On the other hand the approximate continuity alone is not enough to assure the relaxation result, as shown in a counterexample given in [1]. In that

Keywords and phrases. Lower semicontinuity, relaxation, BV-functions, blow-up.

1 Dipartimento di Metodi e Modelli Matematici, via A. Scarpa 16, 00161 Roma, Italy;[email protected];

[email protected]

2 Dipartimento di Matematica e Applicazioni Monte Sant’Angelo, via Cintia, 80126 Napoli, Italy;[email protected]

c EDP Sciences, SMAI 2007

Article published by EDP Sciences and available at http://www.edpsciences.org/cocvor http://dx.doi.org/10.1051/cocv:2007015

paper the representation formula proved here has been obtained in the particular case where the integrand admits a separate dependence on the spatial and the gradient variables. In that case the authors prove the relaxation result by suitably refining the techniques introduced in [5] through the use of a new Reshetnyak-type theorem which applies only to measures which are gradients of BV-function, but does not require the continuity of the integrand.

Unfortunately, this technique does not work for a general integrand like the ones considered here.

Thus, we have to follow another approach to relaxation by using the blow-up technique introduced by Fonseca and M¨uller in [16] and [17] (see also [3]). However, in all these papers the use of this technique relies strongly on the continuity of the integrand with respect to x, an assumption that here is replaced by theH^N−1-a.e. approximate continuity and weak differentiability inx.

This fact introduces some relevant difficulties and requires a delicate study of the approximate continuity of (N−1)-dimensional restrictions of BV-functions. Differently from the usual continuity, the approximate continuity is not inherited by the sections of measurable functions. However, in the first part of this paper we prove that given a H^N−1-almost everywhere approximately continuous BV-function, its sections keep the same property, as long as we restrict them to a countablyH^N−1-rectifiable set whose normal is “never” orthogonal to the hyperplane with respect to whom the sections are taken (see Th. 4.6). This theorem is the main tool needed for dealing with the jump part of the functionalvia the blow-up technique. More precisely, given a jump point x₀ of a BV-functionu, we study the behaviour of the integrand on the tangent hyperplane Π to the jump set atx₀. If the restriction of the integrand to Π is not approximately continuous, we have to approximate Π with a sequence of “good” hyperplanes (where the restriction of f is approximately continuous). In fact, in Proposition 6.5 we prove that this property holds at H^N−1-pointx₀of the jump set ofu.

The paper is organized as follows: Section 2 is devoted to notations; in Section 3 we recall some properties of BV-functions and some results of geometric measure theory needed for the sequel. In Section 4 we carry on a thorough analysis of the fine properties of the (N−1)-dimensional sections of BV-functions. In Section 5 we set the problem and state some technical lemmas; moreover, we discuss some properties of the recession function that do not follow from the corresponding ones of integrand (see Ex. 5.3). Finally, in Section 6 we state and prove our main result, i.e., the relaxation theorem.

2. Notation

Throughout the paper,N ≥2 is a fixed integer and the lettercdenotes a strictly positive constant, whose value may vary from line to line. Given x₀∈IR^N andρ >0,Bρ(x₀) denotes the ball in IR^N centered inx₀ with radius ρ, while S^N−1is the unit sphere of IR^N.

Let Ω be a bounded open set in IR^N. We denote byA(Ω) the family of all bounded open subsetsA of Ω and by B(Ω) the σ-algebra of all Borel subsets B of Ω. Moreover,M(Ω; IR^N) is the space of the IR^N-valued Radon measures on Ω; in particular,M(Ω) :=M(Ω; IR).

As usual,L^N stands for the outer Lebesgue measure on IR^N andH^k for the k-dimensional Hausdorff measure on IR^N. The Lebesgue measure of the unit ball in IR^N is denoted byω_N, henceL^N(Bρ(x₀)) =ω_Nρ^N.

Given a directionν ∈S^N−1, every point x∈IR^N can be decomposed as x= (x^⊥_ν, xν), withxν =x, νν and x^⊥_ν =x−xν. By π_ν⊥ we denote the projection of IR^N onto the plane through the origin orthogonal toν and π_ν denotes the projection of IR^N over the line through the origin in the directionν. We shall often identifyπ_ν⊥(IR^N) with IR^N−1 and π_ν(IR^N) with IR, so that, for instance, the H^N−1-measure on π_ν⊥(IR^N) will be identified with L^N−1. Finally, if E is a given subset of IR^N, we set

Ex^⊥_ν ={xν∈π_ν(E) : (x^⊥_ν, xν)∈E} and Ex_ν ={x^⊥ν ∈π_ν⊥(E) : (x^⊥_ν, xν)∈E}.

Similarly, ifg: IR^N →IR is a given function, for everyx^⊥_ν ∈IR^N−1, we denote bygx^⊥_ν the restriction of the function g to IR;i.e., the functionxν ∈IR→g(x^⊥_ν, xν); for everyxν ∈IR, the restrictiongxν is defined analogously.

Whenν = eN, we simply write π_N−1,π₁, (x, y),Ex,Ey, gx, gy, instead ofπ_e⊥

N,π_eN, (x^⊥_eN, x_eN),Ex^⊥_e

N,Ex_e

N, gx^⊥_e

N,gx_e

N.

3. Basic properties of BV -functions and a coarea formula

Letu∈L¹_loc(Ω); we say thatuhas anapproximate limitatx∈Ω if there existsz∈IR such that

εlim→0⁺−

B_ε(x)|u(y)−z|dx= 0, where −

Bε(x) stands for _LN(B¹ε(x))

Bε(x). Let Su be the set of points where the previous property does not hold, the so-calledapproximate discontinuity set. Note that it is a Borel set. Ifx∈Su,z is uniquely determined, it is called theapproximate limitofuatxand it is denoted byu(x). We recall that u: Ω\Su→IR is a Borel function.

We say thatuisapproximately continuousatxifx∈Suandu(x) =u(x). Clearly, xis a point of approximate continuity of u if and only if is a Lebesgue point of u and since L^N-almost every x ∈ Ω is a Lebesgue point, L^N(Su) = 0. Notice that in general the above definition of approximate continuity is stronger than the usual one given by Federer (see [13]). However, ifu∈L^∞_loc(Ω) the two notions agree.

We say that x₀ ∈Su is an approximate jump pointofuif there exist a, b∈IR and ν ∈S^N−1, such thata=b and

εlim→0⁺−

B_ε⁺(x₀,ν)|u(x)−a|dx= 0, lim

ε→0⁺−

B⁻_ε(x₀,ν)|u(x)−b|dx= 0,

whereB^±_ε(x₀, ν) =x₀+εB_ν^± andB_ν^±={x∈B₁(0) :x, ν≷0}. The triplet (a, b, ν), uniquely determined by the previous definition up to a permutation of a, band a change of sign ofν, is denoted by (u⁺(x₀), u⁻(x₀), νu(x₀)).

We adopt the convention thatu⁺(x₀)> u⁻(x₀). The set of approximate jump points is denoted by Ju and is a Borel set. The quantityu⁺−u⁻ is the jump ofuacross the interfaceJu andνu is the direction of the jump.

The space BV(Ω) is defined as the space of all functions u: Ω →IR belonging to L¹(Ω) whose distributional gradientDuis an IR^N-valued Radon measure (i.e.,Du∈ M(Ω; IR^N)) with total variation|Du|bounded in Ω. We indicate byD^auand D^sutheabsolutely continuousand the singular part of the measureDu with respect to the Lebesgue measure. We recall thatD^auandD^suare mutually singular, moreover we can write

Du=D^au+D^su and D^au=∇uL^N,

where∇uis theRadon-Nikod´ym derivativeofD^auwith respect to the Lebesgue measure. In particular, D^su=D^cu+ (u⁺−u⁻)νuH^N−1 Ju

andJu is acountablyH^N−1-rectifiableBorel set (see [2], Def. 2.57) contained inSu, such thatH^N−1(Su\Ju) = 0.

The remaining partD^cuis called the Cantor partofDu.

Remark 3.1. Since, for everyu∈BV(Ω), Ju is a countably H^N−1-rectifiable set, henceσ-finite with respect to H^N−1, it follows that the set

{ν ∈S^N−1 : H^N−1({x∈Ju : νu(x) =±ν})>0}

is at most countable.

Finally, we define theprecise representativeof a functionu∈∈L¹_loc(Ω) as

u^∗(x) =

⎧⎪

⎨

⎪⎩

u(x) ifx∈Ω\Su u⁺(x)+u⁻(x)

2 ifx∈Ju

0 ifx∈Su\Ju.

Clearly,u^∗: Ω→IR is a Borel function coincidingL^N-almost everywhere withuandu.

Let us recall that if S ⊂IR^N is a countably H^N−1-rectifiable set, then for H^N−1-a.e. x∈S there exists the approximate tangent planeπ_x^StoSatx(see [2], Th. 2.83). A unit vector orthogonal toπ^S_x is called anapproximate

normal to S at xand denoted by ν^S(x). Notice that if S is the jump setJu of some BV function u, then ([2], Th. 3.59)ν^Ju(x) =±νu(x) forH^N−1-a.e. x∈Ju.

Let 1 ≤ k ≤ N−1 be a given integer. By πk,N : IR^N → IR^k we denote the projection of IR^N over the first k components. The formula (3.1) below is a consequence of the general coarea formula for rectifiable set [2], Theorem 2.93, and will be used in the sequel. For the reader’s convenience we give an explicit proof.

Theorem 3.2. LetSbe a countablyH^N−1-rectifiable subset ofIR^N andg: IR^N →[0,+∞]a Borel function. Then, for any integer1≤k≤N−1,

Sg(x) ^N

i=k+1

|ν^S(x), ei|²dH^N−1(x) =

IR^kdt

π_k,N⁻¹(t)∩Sg(x) dH^N−k−1(x). (3.1) Proof. From the coarea formula for rectifiable sets [2], Theorem 2.93 we have that

Sg(x)CkLxdH^N−1(x) =

IR^kdt

π_k,N⁻¹(t)∩Sg(x) dH^N−k−1(x), (3.2) where

CkLx:=

det(Lx◦L^∗_x),

Lx:π^S_x →IR^k is the differential ofπk,N onS atxandL^∗_x is the adjoint of the linear mapLx.

Let us denote by{τ₁, . . . , τN−1}an orthonormal base forπ^S_x; thus,{τ₁, . . . , τN−1, ν^S(x)}is an orthonormal base for IR^N. Denoting by (aij) thek×kmatrix representing the linear mapLx◦L^∗_xwith respect to the standard base in IR^k, we get that

aij=

N−1 h=1

τh, eiτh, ej for alli, j= 1, . . . , k.

Writing, for alli, j,ei=

N−1 h=1

τh, eiτh+ν^S(x), eiν^S(x), we get

δij=ei, ej=

N−1 h=1

τh, eiτh, ej+ν_i^Sν_j^S,

hence

aij =δij−ν_i^Sν_j^S for alli, j= 1, . . . , k.

From this formula it follows that

CkLx=

det (I−ν⊗ν),

where Iis thek×kidentity matrix andν= (ν₁^S(x), . . . , ν_k^S(x)). The assertion then follows by observing that

det (I−ν⊗ν) = 1− |ν|²= N i=k+1

|ν_i^S|². (3.3)

For a general survey on measures and BV-functions we refer to [2, 12, 13, 19, 22].

4. Sections of BV-functions

In this section we state some fine properties of BV-functions, which will be needed in the sequel.

Lemma 4.1. Let g: Ω→IR be a Borel function. Assume thatg∈L¹_loc(Ω)and set G^∗:=

(x, y)∈Ω\Sg : lim

ε→0⁺−

Q(x,ε)|g(z, y)−g^∗(x, y)|dz= 0

,

whereQ(x, ε) :=x+εQ andQ= (−1/2,1/2)^N−1. Then G^∗ is a Borel set.

Proof. Notice thatG^∗=G₁∩G₂, where G₁:=

(x, y)∈Ω\Sg: there exists z∈IR such that lim

ε→0⁺−

Q(x,ε)|g(z, y)−z|dz= 0

and

G₂:=

(x, y)∈Ω\Sg: lim

ε→0⁺−

Q(x,ε)g(z, y) dz=g^∗(x, y)

.

We claim thatG₁ is a Borel set. In fact, consider a dense sequence{qi} ⊂IR and, for everyi, j∈IN, set Gij =

(x, y)∈Ω : lim sup

ε→0⁺ −

Q(x,ε)|g(z, y)−qi|dz< 1 j

.

It is not difficult to check that if h : IR^N → IR is a locally summable Borel function, then also (x, y) →

−

Q(x,ε)h(z, y)dz is a Borel function for any ε > 0. From this fact we get immediately that each Gij is a Borel set, and since

G₁= ∞ j=1

∞ i=1

Gij,

our claim follows. On the other hand, since for any ε >0 the function (x, y)→ −

Q(x,ε)g(z, y)dz is Borel, we

easily get that alsoG₂ is a Borel set. This concludes the proof.

Lemma 4.2. ([2], Th. 3.108) Let g ∈BV(Ω) be a given function. Then, forL^N−1-almost every x ∈π_N−1(Ω), the functiong_x belongs to BV(Ω_x). Moreover J_g

x = (J_g)_x and(g^∗)_x(y) = (g_x)^∗(y)for every y∈Ωx \(J_g)_x. In particular, if (J_g)_x =∅, then (g^∗)_x(y) = (g_x)^∗(y)for every y∈Ωx and both functions (g^∗)_x and(g_x)^∗ are continuous in Ωx.

Lemma 4.3. Letg∈BV(Ω)be a given function. Then, forL¹-almost everyy∈π₁(Ω), the functiongy belongs to BV(Ωy).

Proof. IfN = 2, the property follows by Lemma 4.2. IfN >2, the property can be easily obtained following the

proof of [2], Theorem 3.103.

In the next two lemmas we assumeN >2. For everyx= (x, y)∈IR^N, we set ˆ

x_i = (x₁, . . . , xi−1, xi+1, . . . , xN−1)∈IR^N−2

and write, for the sake of simplicity,x= (xi,ˆx_i, y). Moreover, ifE is a given set, withEx_ˆiy we denoteE_x⊥ν, with ν = ei; a similar notation will be used also for functions.

Lemma 4.4. Let g ∈ BV(Ω) be a given function. Then there exists a set N₀ ⊂ IR with L¹(N₀) = 0 with the following property: for every y ∈ IR\N₀, gy ∈ BV(Ωy) and, for every i = 1, . . . , N −1, there exists a set N^iy ⊂IR^N−2 with L^N−2(N^iy) = 0such that, for every ˆx_i∈IR^N−2\N^iy we have thatg_ˆx_iy∈BV(Ωx_ˆiy)and

(S_gy)_xˆ i =J_g

xˆi y

(4.1) [(g_y)^∗]_xˆ

i(xi) = (g_ˆx

iy)^∗(xi) for every xi∈Ωx_ˆiy\J_g

ˆ x

i y

. (4.2)

Proof. For the sake of simplicity, assume Ω = IR^N. By Lemma 4.3 there existsN₀⊂IR withL¹(N₀) = 0 such that for everyy∈IR\N₀we have thatg_y∈BV(IR^N−1),J_gy is a countablyH^N−2-rectifiable set andH^N−2(S_gy\Jg_y) = 0.

Moreover, by Theorem 3.2, we have that, for every i= 1, . . . , N−1, 0 =

S_gy\J_gy|ν^Sgy, ei|dH^N−2=

IR^N−2H⁰((S_gy \J_gy)_ˆx i) dˆx_i

so that there exists a set N₁^iy ⊂ IR^N−2 with L^N−2(N₁^iy) = 0, such that for every ˆx_i ∈ IR^N−2 \N₁^iy we have (S_gy \J_gy)_ˆx

i=∅;i.e.,

(Sg_y)_ˆx

i = (J_gy)_ˆx

i. (4.3)

Moreover, by Lemma 4.2 applied to gy, we have that for everyy ∈IR\N₀ and for everyi= 1, . . . , N −1, there exists a set N₂^iy ⊂IR^N−2 withL^N−2(N₂^iy) = 0, such that for every ˆx_i∈IR^N−2\N₂^iy we have thatg_xˆ

iy∈BV(IR) and

(Jgy)_ˆx i =J_g

xˆiy

. (4.4)

[(g_y)^∗]_xˆ

i(xi) = (g_xˆ

iy)^∗(xi) for everyxi∈IR\J_g

ˆ x

i y

. (4.5)

Finally, for everyy∈IR\N₀and for everyi= 1, . . . , N−1, setN^iy=N₁^iy∪N₂^iy ⊂IR^N−2, so thatL^N−2(N^iy) = 0 and for every ˆx_i∈IR^N−2\N^iy we have that (4.3), (4.4) and (4.5) hold. Hence the assertion follows.

Lemma 4.5. Let g ∈BV(Ω)be a function which is approximately continuous inH^N−1-almost every point ofΩ.

Then there exists a set M₀⊂IRwith L¹(M₀) = 0with the following property: for every y∈IR\M₀,gy ∈BV(Ωy) and, for every i = 1, . . . , N −1, there exists a set M^iy ⊂ IR^N−2 with L^N−2(M^iy) = 0 such that, for every ˆ

x_i∈IR^N−2\M^iy we have thatgx_ˆiy ∈BV(Ωx_ˆiy)and J_g

ˆxi y

= (S_g)_xˆ

iy= (J_g)_xˆ

iy =∅, (4.6)

(g_xˆ

iy)^∗(xi) = (g^∗)x_ˆiy(xi) for everyxi∈Ωx_ˆiy (4.7) and both functions are continuous.

Proof. As before, we assume for simplicity Ω = IR^N. By assumption we have that H^N−1(Sg) = 0, thus from Theorem 3.2 we get that for everyi= 1, . . . , N−1

0 =

S_g|ν^Sg, ei|dH^N−1=

IR^N−1H⁰((S_g)x_ˆiy) dˆx_i dy.

Therefore there exists M₁ⁱ ⊂ IR^N−1, with L^N−1(M₁ⁱ) = 0, such that for every (ˆx_i, y) ∈ IR^N−1 \M₁ⁱ we have (S_g)_xˆ

iy=∅. Set

M₁={y∈IR :L^N−2((M₁ⁱ)y)>0 for at least one index i= 1, . . . , N−1}.

Then, by Fubini’s theorem

0 =L^N−1(M₁ⁱ) =

IRL^N−2((M₁ⁱ)y) dy,

so that L^N−2((M₁ⁱ)y) = 0 for a.e. y ∈ IR; i.e., L¹(M₁) = 0. Moreover, for every y ∈ IR\M₁ and every ˆ

x_i∈IR^N−2\(M₁ⁱ)y, recalling that (J_g)_ˆx

iy ⊆(S_g)_xˆ

iy, we have (S_g)_xˆ

iy= (J_g)_ˆx

iy =∅. (4.8)

By Lemma 4.2 we have that for everyi= 1, . . . , N−1, there exists a setM₂ⁱ ⊂IR^N−1 withL^N−1(M₂ⁱ) = 0, such that for every (ˆx_i, y)∈IR^N−1\M₂ⁱ we have thatg_ˆx

iy∈BV(IR) and Jg

ˆxiy = (J_g)_xˆ

iy, (4.9)

(g^∗)x_ˆiy(xi) = (g_ˆx

iy)^∗(xi) for everyxi∈IR\(J_g)_xˆ

iy. (4.10)

Set

M₂={y∈IR :L^N−2((M₂ⁱ)y)>0 for at least one index i= 1, . . . , N−1}.

As before, Fubini’s theorem implies that L¹(M₂) = 0. Thus for anyy∈IR\M₂ and any ˆx_i ∈IR^N−2\(M₂ⁱ)y we obtain that (4.9) and (4.10) hold. Finally, setM₀=M₁∪M₂⊂IR andM^iy = (M₁ⁱ)y∪(M₂ⁱ)y, so thatL¹(M₀) = 0 and, for everyy∈IR\M₀,L^N−2(M^iy) = 0. Moreover, for everyy∈IR\M₀, by (4.8), (4.9) and (4.10) we have

∅= (Sg)_ˆx

iy= (Jg)_xˆ

iy=Jg_xˆiy,

(g^∗)x_ˆiy(xi) = (gx_ˆiy)^∗(xi) for everyxi∈IR

and by Lemma 4.2 both functions are continuous.

Next theorem states that given aH^N−1-a.e. approximately continuousBV functiong, its (N−1)-dimensional sections are still H^N−1-a.e. approximately continuous along a countablyH^N−1-rectifiable set whose normals are

“never” orthogonal to the direction in which the sections are taken.

Theorem 4.6. Let g∈BV(Ω)be a Borel function which is approximately continuous in H^N−1-almost every point of Ω. Set

G=

(x, y)∈Ω\Sg: lim

ε→0⁺−

Q(x,ε)|g(z, y)−g(x, y)|dz= 0

.

Let S ⊂ Ω be a countably H^N−1-rectifiable set such that H^N−1({x ∈ S : ν^S(x) = ±eN}) = 0. Then H^N−1(S\G) = 0.

Proof. Again, we assume for simplicity that Ω = IR^N.

Sinceg is approximately continuousH^N−1-a.e., the thesis will be achieved if we prove thatH^N−1(S\G^∗) = 0, whereG^∗ is the set defined in Lemma 4.1. To this aim, let us first assume thatN >2.

Following the notation used in Lemmas 4.4 and 4.5 let us takey∈IR\(N₀∪M₀) and for everyi= 1, . . . , N−1, ˆ

x_i∈IR^N−2\(N^iy∪M^iy). Then by (4.1) and (4.6) we obtain (Sg)_xˆ

iy = (Sg_y)_xˆ i =∅

so that, for everyxi ∈IR we have thatxi∈(Sg)x_ˆiy, (i.e.,x= (xi,xˆ_i, y)∈Sg) andxi ∈(Sgy)x_ˆi (i.e.,x= (xi,xˆ_i)∈ Sgy). Hence, xis a point whereg has an approximate limit andx is a point wheregy has an approximate limit.

Moreover by (4.2) and (4.7) it follows that (gy)^∗(x) = (g^∗)_ˆx_iy(xi) =g^∗(x),i.e.,

εlim→0⁺−

Q(x,ε)|g(z, y)−g^∗(x)|dz= lim

ε→0⁺−

Q(x,ε)|g(z, y)−(gy)^∗(x)|dz = 0, which implies that x ∈ G^∗; i.e., xi ∈ G^∗_xˆ

iy. In particular we obtain that, for every y ∈ IR\(N₀∪M₀), every i= 1, . . . , N−1 and every ˆx_i ∈IR^N−2\(N^iy∪M^iy)

G^∗_ˆx

iy = IR. (4.11)

Since|ν^Sy|= 1, from Theorem 3.2 and (4.11) we obtain H^N−2((S\G^∗)y) =

N−1 i=1

(S\G^∗)y

|ν^Sy, ei|²dH^N−2≤

N−1 i=1

(S\G^∗)y

|ν^Sy, ei|dH^N−2

=

N−1 i=1

IR^N−2H⁰([(S\G^∗)y]x_ˆi) dˆx_i=

N−1 i=1

IR^N−2H⁰((S\G^∗)_ˆx_iy) dˆx_i= 0, which impliesH^N−2((S\G^∗)y) = 0 for everyy∈IR\(N₀∪M₀). Finally, using again Theorem 3.2, we obtain

S\G^∗

1− |ν^S, eN|²dH^N−1=

IRH^N−2((S\G^∗)y) dy= 0.

Therefore, taking into account the assumption made onS, we haveH^N−1(S\G^∗) = 0.

IfN = 2, we apply the coarea formula (3.1) again, thus getting

S\G^∗|ν^S, e₁|dH¹=

IRH⁰((S\G^∗)y) dy= 0,

where the last equality holds sinceH¹(Sg) = 0 implies (Jg)y =∅forL¹-almost everyy∈π₁(Ω) and, by Lemma 4.2, (g^∗)y(x) = (gy)^∗(x) for allx∈Ωy and forL¹-almost everyy∈π₁(Ω). Hence, the assertion follows.

Remark 4.7. Clearly, Theorem 4.6 still holds if we replace eN by a generic directionν. More precisely, given any direction ν∈S^N−1, set

Gν =

x= (x^⊥_ν, xν)∈Ω\Sg: lim

ε→0⁺−

Q^⊥_ν(x,ε)|g(zν^⊥, xν)−g(x^⊥_ν, xν)|dz^⊥_ν = 0

,

where Q^⊥_ν(x, ε) = π_ν⊥(x+εQν), Qν = Rν(−1/2,1/2)^N and Rν denotes a rotation such that RνeN =ν. Then H^N−1(S\Gν) = 0 for every countablyH^N−1-rectifiable subsetSof Ω, such thatH^N−1({x∈S:ν^S(x) =±ν}) = 0.

5. Setting of the problem

Letf : Ω×IR×IR^N →IR be a Borel function satisfying the following conditions:

⎧⎪

⎪⎪

⎪⎪

⎨

⎪⎪

⎪⎪

⎪⎩

(i) f(·, s, ξ)∈W^1,1(Ω),for every (s, ξ)∈IR×IR^N;

(ii) f(·, s, ξ) is approximately continuousH^N−1-a.e.in Ω,for every (s, ξ)∈IR×IR^N; (iii) for every bounded set B⊂IR×IR^N there exists a constantL(B) such that

Ω|∇xf|(x, s, ξ) dx < L(B) ∀(s, ξ)∈B.

(5.1)

Remark 5.1. Notice that assumption (ii) of (5.1) seems redundant, since everyW^1,1-function admits aH^N−1-a.e.

approximately continuous representative. Moreover, the functional in (5.5) is clearly not affected by the choice of the representative. However, functional (5.6) does depend on the particular representative chosen. Therefore, the representation formula provided by Theorem 6.1 below does not hold if we take a representative of f not satisfying (ii).

We will assume that

f(x, s,·) is convex for every (x, s)∈Ω×IR; (5.2)

|f(x, s, ξ)−f(x, s₀, ξ)| ≤Λ 1 +|ξ|

|s−s₀| for every (x, s, ξ),(x, s₀, ξ)∈Ω×IR×IR^N; (5.3) 0≤f(x, s, ξ)≤Λ(1 +|ξ|) for every (x, s, ξ)∈Ω×IR×IR^N, (5.4)

for some positive Λ. From (5.2) and (5.4), it follows thatf is Lipschitz continuous in the last variable, uniformly with respect to (x, s).

For everyA∈ A(Ω) and everyu∈BV(Ω), we define

F(u, A) =

⎧⎪

⎪⎨

⎪⎪

⎩

Af(x, u,∇u) dx ifu∈W^1,1(Ω)

+∞ ifu∈BV(Ω)\W^1,1(Ω).

(5.5)

Our aim is to prove an integral representation theorem for the relaxationF ofF, with respect to theL¹-topology.

We recall that the relaxation ofF is the greatest lower semicontinuous functional not greater thanF;i.e., F(u,Ω) := inf{lim inf

n→+∞F(un,Ω) : un ∈W^1,1(Ω), un →uin L¹(Ω)}. Among the main properties of the relaxation, we recall the following ones:

(i) for everyA∈ A(Ω),F(·, A) is lower semicontinuous with respect to theL¹-topology;

(ii) for everyA∈ A(Ω),F(·, A) is local;i.e., for everyu, v∈BV(Ω), withu=vonA,F(u, A) =F(v, A);

(iii) for everyu∈BV(Ω), F(u,·) is aσ-additive measure onB(Ω).

For other properties of the relaxation we refer to [4, 6, 10, 11].

We set, for everyA∈ A(Ω) and everyu∈BV(Ω), F(u, A) =

Af(x, u,∇u) dx+

Af^∞

x,u, D^cu

|D^cu|

d|D^cu|+

Ju∩A

u⁺(x)

u⁻(x)f^∞(x, s, νu) ds

dH^N−1(x), (5.6)

wheref^∞: Ω×IR×IR^N →IR is the so-calledrecession function off, defined by f^∞(x, s, ξ) = lim

t→+∞

f(x, s, tξ)

t = sup

t>0

f(x, s, tξ)−f(x, s,0)

t · (5.7)

Notice that assumptions (5.2) and (5.4) imply that the limit in (5.7) exists for every (x, s, ξ)∈Ω×IR×IR^N (since the function t→ ^{f(x,s,tξ)−}_t^f(x,s,0) is increasing). Moreover, the function f^∞is convex and positively homogeneous of degree one in the last variable and, as a consequence of definition (5.7), we have that

f(x, s, tξ)

t ≤f^∞(x, s, ξ) +f(x, s,0)

t for allt >0. (5.8)

Notice also that f^∞ is a Borel function in Ω×IR×IR^N. Thus, the functional F in (5.6) is well defined. By the assumptions made onf, it follows that

0≤f^∞(x, s, ξ)≤Λ|ξ| for every (x, s, ξ)∈Ω×IR×IR^N, (5.9)

|f^∞(x, s, ξ)−f^∞(x, s₀, ξ)| ≤Λ|ξ||s−s₀| for every (x, s, ξ),(x, s₀, ξ)∈Ω×IR×IR^N. (5.10) In the sequel, we will assume also that

f^∞(·, s, ξ)∈BV(Ω) for every (s, ξ)∈IR×IR^N, (5.11) and that, for any (s, ξ)∈IR×IR^N,

f^∞(·, s, ξ) is approximately continuous forH^N−1-a.e. x∈Ω. (5.12)