On L1-Lower Semicontinuity in BV

Journal of Convex Analysis Volume 12 (2005), No. 1, 173–185

Virginia De Cicco Universitàdi Roma, Dipartimento di Metodi e Modelli Matematici per le Scienze Applicate, Via Scarpa 16, 00161 Roma, Italy [email protected] Nicola Fusco Universitàdi Napoli, Dipartimento di Matematica e Applicazioni, Via Cinzia, 80126 Napoli, Italy [email protected] Anna Verde Universitàdi Napoli, Dipartimento di Matematica e Applicazioni, Via Cinzia, 80126 Napoli, Italy [email protected]

Received July 18, 2003

A lower semicontinuity result is obtained for the BV extension of an integral functional of the type

f(x, u(x), u(x)) dx , ∇ ZΩ where the energy density f is not coercive and satisﬁes mild regularity assumptions.

1. Introduction Since the celebrated paper [20] of Serrin appeared in 1961, many authors have contributed to the study of the L1-lower semicontinuity of a functional of the type

f(x; u(x); u(x)) dx ; u W 1,1(Ω); (1) ∇ ∈ ZΩ and of the corresponding extension (3) to BV (Ω), with the aim of understanding which are the minimal assumptions on f that guarantee lower semicontinuity. Here, the starting point is the result, proved in [20], stating that if the integrand f(x; u; ‚) is a nonnegative, continuous function from Ω IR IRN , convex in ‚, and such that the × × (classical) derivatives xf; ξf; x ξf exist and are continuous functions, then the integral functional (1) is lower∇ semicontinuous∇ ∇ ∇ in W 1,1(Ω), with respect to the L1 convergence in the open set Ω. In 1983 De Giorgi, Buttazzo and Dal Maso (see [10]) showed that Serrin’s assumptions can be substantially weakened when dealing with autonomous (i.e. f f(u; ‚)) functionals of the type (1). In this case they were able to prove the L1-lower semicontinuity≡ without even assuming f to be continuous in u for all ‚ IRN . Their result is proved by approximating the integrand f with a sequence of aﬃne∈ functions and then using a suitable version of the chain rule in the Sobolev space W 1,1(Ω) in order to get the lower semicontinuity of the approximating functionals.

ISSN 0944-6532 / $ 2.50 c Heldermann Verlag 174 V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV

On the other hand, things are more complicate if we allow f to depend explicitly on x, since convexity in ‚ and continuity with respect to all variables (x; u; ‚) are not enough to ensure the lower semicontinuity of functional (1), as shown by a well known example of Aronszajin (see [19]). An even more surprising counterexample to lower semicontinuity has been recently given by Gori, Maggi and Marcellini in [18] (see also [17]). In their example, which is one dimensional, the integrand f is equal to a(x; u)‚ 1 and the function a is Höldercontinuous in x, uniformly with respect to u. | − | These examples clearly show that in order to get lower semicontinuity we must retain, in the spirit of Serrin’s theorem recalled above, some differentiability of f with respect to x. This is done, for instance, by Gori, Maggi and Marcellini in [18]. They prove that if f is continuous in (x; u; ‚), convex in ‚ and weakly differentiable in x and if for any open set N Ω0 Ω and any bounded set B IR IR there exists a constant L(Ω0;B) such that ⊂⊂ ⊂ ×

xf(x; u; ‚) dx L(Ω0;B) for every (u; ‚) B; (2) Ω ∇ ≤ ∈ Z 0 ¬ ¬ ¬ ¬ then the functional¬ (1) is L1-lower¬ semicontinuous. Also this result is proved by approximating the integrand f with a sequence of affine functions whose coefficients are explicitly given in terms of f (see Lemma 2.1 below, proved in [9]). The lower semicontinuity result by Gori, Maggi and Marcellini has been extended in a later paper ([8]) by De Cicco and Leoni, who prove a fairly general version of the chain rule for vectors fields with divergence in L1. Using this chain rule, they get the lower semicontinuity of the functional (1) under the assumption that f is continuous in u and convex in ‚ and that the distributional gradient ξf satisfies a suitable assumption. Namely, they require that, for 1-a.e. u IR and ∇N -a.e. ‚ IRN , f( ; u; ‚) is locally L ∈ L ∈ ∇ξ · summable in Ω, its distributional divergence divx ξf(x; u; ‚) is locally summable in Ω f x; u; ‚ L1 N ∇ and divx ξ ( ) belongs to loc(Ω IR IR ). These assumptions seem very close to being∇ necessary conditions for lower× semicontinuity× (see, for instance, [14], [16], [3] in the BV setting, and [8] for a discussion on this subject).

In this paper, we prove a lower semicontinuity result for the functional

Dcu u+(x) c N 1 F (u; Ω) = f(x; u; u)dx + f ∞(x; u; )d D u + d − f ∞(x; t; • )dt; ∇ Dcu | | H u ZΩ ZΩ | | ZJu Zu−(x) (3) e where u BV (Ω), u is the approximate limit of u, f ∞ denotes the recession function of f with respect∈ to ‚ and f(x; u; ‚) is a suitable Borel function, continuous in u, convex in ‚ u; ‚ N f ; u; ‚ N 1 and such that fore any ( ) IR IR the function ( ) coincides − -a.e. in Ω with the precise representative∈ of f(×; u; ‚) (see the deﬁnition· given in (5)).H As usual, Dcu denotes the Cantor part of the measure· Du, Dcu= Dcu is the derivative of the measure c c | | D u with respect to its total variation D u and Ju is the jump set of u (the deﬁnitions of all these quantities are recalled in Section| | 2). The functional F is the natural extension of the functional (1) to the space BV (Ω). In fact (see, for instance, the paper [5] by Dal Maso), under standard continuity and coercivity assumptions on f, F coincides with the relaxed functional of (1), that is the greatest L1-lower semicontinuous functional on BV (Ω) coinciding with (1) on W 1,1(Ω). V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV 175

Therefore, the study of the lower semicontinuity of the functional F is essentially parallel to the Sobolev case, though occasionally things may turn out to be more difficult to prove in the BV setting. Among the many papers devoted to the BV case we just mention here the ones which are closer to our approach, such as [6], [7], [4], [12], [13] and [15], where the lower semicontinuity result of [18] is extended to the functional F under the extra assumption that f is continuous. Here we give a further improvement of the result of [15] by dropping the continuity of f with respect to x. Namely we prove the following result. Theorem 1.1. Let Ω be an open subset of IRN and f :Ω IR IRN [0; + ) a locally bounded Carathéodory function, such that, for every (u;× ‚) IR× IRN→, the function∞ f( ; u; ‚) is weakly differentiable in Ω. Let us assume that there exists∈ × a set Z Ω, with N·(Z) = 0, such that ⊂ L (i ) f(x; u; ) is convex in IRN for every (x; u) (Ω Z) IR; · ∈ \ × (ii ) f(x; ;‚) is continuous in IR for every (x; ‚) (Ω Z) IRN · ∈ \ × N and that for any open set Ω0 Ω and any bounded set B IR IR the estimate (2) holds. Then, the functional⊂⊂F defined in (3) is lower semicontinuous⊂ × in BV (Ω) with respect to the L1(Ω) convergence.

Beside the usual approximation from below of the integrand f by a sequence of affine functions, the proof of the above theorem uses a localization argument which allows us to prove separately the lower semicontinuity of the diffuse part and of the jump part of the functional F . However, the main technical tool used in the proof is provided by the integration by parts formula stated below. This formula does not seem to be contained in any of the similar results known in the literature and we think that could be of independent interest. Proposition 1.2. Let b : IRN IR IR be a bounded Borel function with compact support in IRN IR, such that, for any×t →IR, b( ; t) is weakly differentiable in IRN . Assume that × ∈ · (j ) there exists a set Z IRN , with N (Z) = 0, such that for every x Ω Z the function b(x; ) is continuous⊂ in IR;L ∈ \ · (jj ) there exists a constant L such that for any t IR ∈ @b x; t dx L: @x( ) IRN ≤ Z ¬ ¬ ¬ ¬ Then, for every u BV (IRN ) and for every¬ C¬ 1(IRN ), we have ∈ ∈ 0 u(x) u(x) @b (x)dx b(x; t)dt = (x)dx (x; t) dt N ∇ − N @x ZIR Z0 ZIR Z0 (x)b(x; u (x)) u(x) dx (x) b(x; u (x))dDcu − N ∇ − N ZIR ZIR u+(x) N 1 e (x)• (x)d − b(x; t)dt ; − u H ZJu Zu−(x) where b is defined as in (5). 176 V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV

2. Definitions and preliminary results In this section we recall a few preliminary results needed in the sequel and some basic definitions of the theory of BV functions. We follow the notation used in [2] and we refer to this book for all the properties of BV functions used here. The first lemma is an approximation result due to De Giorgi (see [9]).

fi C N fi fi d Lemma 2.1. There exists a sequence j in 0∞(IR ), with j 0 and j( ) = 1, ≥ N ZIR with the property that, whenever f :Ω IR IRN [0; + ) is a Carathéodory function, if we set, for any j IN and any x Ω×; u ×IR;‚→ IRN , ∞ ∈ ∈ ∈ ∈

a0,j(x; u) = f(x; u; ) ((N + 1)fij() + fij(); ) d ; N h∇ i ZIR @fij ai,j(x; u) = f(x; u; ) () d; for i = 1;:::;N; (4) − N @ ZIR i N

gj(x; t; ‚) = a0,j(x; u) + ai,j(x; u)‚i ; i=1 X then, for all (x; u) Ω IR such that f(x; u; ) is convex, we have ∈ × · N f(x; u; ‚) = sup max gj(x; u; ‚); 0 for all ‚ IR : j IN { } ∈ ∈

Notice that if f is a Carathéodory (resp. Borel) function, then also the ai,j are Carathéodo- ry (resp. Borel) functions in Ω IR and if f satisfies (2) then a similar estimate is also × satisfied by the functions ai,j (locally uniformly with respect to u). Let k :Ω IRM IR be a locally bounded Carathéodory function. For any x Ω, z IRM , we× set → ∈ ∈ k(x; z) = lim sup k(y; z) dy : (5) r 0 −B (x) → Z r Notice that k is a Borel function and that k(x; ) is a continuous function for any x Ω. · ∈ N N Let us recall that if h : IR IR is a convex function, its recession function h∞ : IR IR is defined by setting → → h t‚ ( ) N h∞(‚) = lim for any ‚ IR : (6) t + t → ∞ ∈ If h : V IRN IR is a any function, such that for any v in the set V , the function h(v; ) × → · is convex, we denote by h∞(v; ‚) the recession function of h(v; ). The following result is an easy consequence of the definition (6) and of Lemma 2.1 (see· also [2, Lemma 2.33]). Lemma 2.2. Let f :Ω IR IRN [0; ) be a Carathéodory function, satisfying the assumptions of Theorem×1:1.× Then,→ for all∞(x; u; ‚) Ω IR IRN , ∈ × ×

f ∞(x; u; ‚) = sup max «j(x; u);‚ ; 0 ; j IN {h i } ∈ V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV 177

where for any i = 1;:::;N,

@ﬁj «i,j(x; u) = f(x; u; ) () d : (7) − N @ ZIR i

An immediate consequence of this lemma is that the function (x; u; ‚) f ∞(x; u; ‚) is a Borel function. Thus, the functional F in (3) is well deﬁned. 7→ The following result is contained in [2, Lemma 2.35]. N Lemma 2.3. Let ¶ be a positive Radon measure in an open set Ω IR and let j : Ω [0; ], j IN, be Borel functions. Then ⊂ → ∞ ∈

sup j d¶ = sup j d¶ ; j Ω º j J Aj » Z X∈ Z

where the supremum ranges among all ﬁnite sets J IN and all families Aj j IN of ∈ pairwise disjoint open sets with compact closure in Ω. ⊂ { }

u L1 u Let be a function in loc(Ω). We say that is approximately continuous at the point x Ω if there exists u(x) IR such that ∈ ∈

e lim u(y) u(x) dy = 0 ; r 0 − | − | → ZBr(x) the value u(x) is called the approximate limiteof u at x. In the sequel, whenever k : Ω IRM IR is a locally bounded Carath´eodory function, we shall denote by k(x; u) the × → approximatee limit of k( ; u) at x, provided that this limit exists; in this case we have also · k(x; z) = k(x; z). The set Cu of all points where u is approximately continuouse is a Borel set. We say that a point x Ω Cu is an approximate jump point for u if there exist + ∈ N\1 + eu (x); u−(x) IR and • (x) S − such that u−(x) < u (x) and ∈ u ∈

+ lim u(y) u (x) dy = 0; lim u(y) u−(x) dy = 0 ; r 0 − + | − | r 0 − | − | → ZBr (x;νu(x)) → ZBr−(x;νu(x))

+ where B (x; • (x)) = y B (x): y x; • (x) > 0 and B−(x; • (x)) is defined r u { ∈ r h − u i } r u analogously. Also the set Ju Ω Cu of all approximate jump points is a Borel set and + ⊂ \ N 1 the function (u (x); u−(x);• (x)) : J IR IR S − is a Borel function. u u → × × Given a point x Cu, we say that u is approximately differentiable at x if there exists u(x) IRN such∈ that ∇ ∈ 1 lim u(y) u(x) u(x); y x dy = 0 : r 0 rN+1 | − − h∇ − i| → ZBr(x) The vector u(x) is called the approximatee differential of u at x. The set of points in ∇ Cu where the approximate differential of u exists is a Borel set denoted by u. It can be easily verified that u : IRN is a Borel map. D ∇ Du → 178 V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV

A function u L1(Ω) is called of bounded variation if its distributional gradient Du is an N ∈ IR -vector valued measure and the total variation Du of Du is finite in Ω. The space of all functions of bounded variation in Ω is denoted| by |BV (Ω). If u BV (Ω), we denote by Dau the absolutely continuous part of Du with respect to the Lebesgue∈ measure N . The singular part Dsu can be split in two more parts, the jump part Dju and the CantorL part Dcu, defined by Dju = Dsu J ;Dcu = Dsu Dju : u − Furthermore, a N c j + N 1 D u = u ;D u = Du (C );D u = (u u−)• − J ; ∇ L Du u \Du − uH u N 1 N where − denotes the (N 1)-dimensional Hausdorff measure in IR (see [2, Proposi- tion 3.92]).H We shall also use− the following form of the coarea formula for BV functions (see [11, Theorem 4.5.9])

+ ∞ N 1 g d Du = dt g d − ; (8) Ω | | u t u+ H Z Z−∞ Z{ −≤ ≤ } where g :Ω [0; + ] is a Borel function. → ∞ We conclude this section by proving the following lemma that will be useful in the sequel. Lemma 2.4. Let f be a Carath´eodory function satisfying the assumptions of Theorem 1:1. Let us denote by aj the functions obtained from f as in (5) and by «j those obtained in the same way from f as in (7). Then, for any u IR, there exists a Borel set Zu Ω, N 1 ∈ ⊂ with − (Z ) = 0, such that for any j IN, H u ∈ a (x; u) = a (x; u) = « (x; u) for all x Ω Z : j j j ∈ \ u

Proof. Let us ﬁx u IR. We claim that there exists a Borel set Zu Ω, with N 1 ∈ e N ⊂ − (Zu) = 0, such that, for any x Ω Zu and any ‚ IR , x is a point of approximate continuityH for the function f( ; u; ‚),∈ hence\ in particular∈ · f(x; u; ‚) = f(x; u; ‚) for any x Ω Z and any ‚ IRN : (9) ∈ \ u ∈ To this aim, let us consider a sequence ‚ , dense in IRN , and recall that since the functions e h f( ; u; ‚h) are weakly integrable in Ω, for any h there exists a Borel set Bu,h Ω such · N 1 ⊂ that − (Bu,h) = 0 and x is a point of approximate continuity for f( ; u; ‚h) for any x HΩ B . Let us set Z = B and ﬁx x Ω Z , ‚ IRN .· Let ‚ be a ∈ \ u,h u ∪h u,h ∈ \ u ∈ kh subsequence of ‚h converging to ‚. Since f( ; u; ) is locally bounded and convex in ‚ for N · · N -a.e. x Ω , then for any open set Ω0 Ω and any compact set K in IR , there existsL a constant∈ L such that ⊂⊂ N f(y; u; ‚) f(y; u; ‚0) L ‚ ‚0 for -a.e. y Ω0 and any ‚;‚0 K: | − | ≤ | − | L ∈ ∈ From this inequality, we get that

f y; u; ‚ f x; u; ‚ dy lim sup ( ) ( kh ) r 0 −B (x)| − | → Z r f y; u; ‚ fe y; u; ‚ dy f y; u; ‚ f x; u; ‚ dy lim sup ( ) ( kh ) + lim sup ( kh ) ( kh ) ≤ r 0 −B (x)| − | r 0 −B (x)| − | → Z r → Z r c ‚ ‚ ; k e ≤ | − h | V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV 179 f x; u; ‚ hence we easily deduce that the sequence ( kh ) converges to the approximate limit of f( ; u; ‚) at x and thus that x is a point of approximate continuity for f( ; u; ‚). · · To conclude the proof, we now show that xe is a point of approximate continuity also for the functions a ( ; u), thus obtaining that a (x; u) = a (x; u), and that indeed a (x; u) = j · j j j «j(x; u). In fact, using (9) and Fubini’s theorem, we get e e

aj(y; u) «j(x; u) dy dy [f(y; u; ) f(x; u; )] fij() d −B (x)| − | ≤ −B (x) IRN − ∇ Z r Z r ¬Z ¬ ¬ ¬ = dy¬ [f(y; u; ) f(x; u; )] fij() d¬ −B (x) IRN − ∇ Z r ¬Z ¬ ¬ e ¬ fij¬() d f(y; u; ) f(x; u; ) dy:¬ ≤ N |∇ | − | − | ZIR ZBr(x) e Then, the assertion follows from the Lebesgue dominated convergence theorem since, for any IRN , ∈ lim f(y; u; ) f(x; u; ) dy = 0 : r 0 − | − | → ZBr(x) e

3. Proof of the Theorem 1.1 In this section we give the proof of both Proposition 1.2 and Theorem 1.1. N Proof of Proposition 1.2. Let %ε(x) = "− %(x="), " > 0, a family of molliﬁers, where % is a nonnegative smooth function, with compact support in the unit ball B1, such that %(y)dy = 1. For any " > 0; (x; t) IRN IR, we set N ∈ × ZIR

t bε(x; t) = %ε(x z)b(z; t) dz; fε(x; t) = bε(x; …) d… : N − ZIR Z0

We claim that f C1(IRN IR). To this aim, let B be an open ball such that the support ε ∈ 0 × R of b is contained in BR ( R;R); then the support of fε is contained in BR+ε ( R;R) and thus to prove the× claim− it is enough to show that f C1(B IR).× Let− su fix ε ∈ R+ε × » > 0. Since b is a Carathéodory function in BR+2ε IR, by Scorza Dragoni’s theorem K N B ×K < » b there exists a compact set such that ( R+2ε ) and K IR is continuous. | × Moreover, let fi (0;») be such that L \ ∈

(z; t ); (z; t ) K IR; t t < ﬁ = b(z; t ) b(z; t ) < » : 1 2 ∈ × | 1 − 2| ⇒ | 1 − 2 |

Let us now ﬁx (x ; t ); (x ; t ) B IR such that (x ; t ) (x ; t ) < ﬁ. Then, setting 1 1 2 2 ∈ R+ε × | 1 1 − 2 2 | 180 V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV

M = sup b , we have | | @f @f ε (x ; t ) ε (x ; t ) = % (x z)b(z; t ) dz % (x z)b(z; t ) dz @t 1 1 − @t 2 2 ε 1 − 1 − ε 2 − 2 ¬ ¬ ¬ZBε(x1) ZBε(x2) ¬ ¬ ¬ ¬ ¬ ¬ % x z % x ¬ z¬ b z; t dz % x z b z; t dz ¬ ¬ ε( 1 ) ε( 2¬ ¬) ( 1) + ε( 2 ) ( 1) ¬ ≤ Bε(x )| − − − || | Bε(x ) Bε(x ) − | | Z 1 Z 1 \ 2

+ %ε(x2 z) b(z; t2) dz + %ε(x2 z) b(z; t1) b(z; t2) dz Bε(x ) Bε(x ) − | | Bε(x ) Bε(x ) − | − | Z 2 \ 1 Z 1 ∩ 2

Mcε x1 x2 + %ε(x2 z) b(z; t1) b(z; t2) dz ≤ | − | Bε(x ) Bε(x ) K − | − | Z 1 ∩ 2 \

+ %ε(x2 z) b(z; t1) b(z; t2) dz Bε(x ) Bε(x ) K − | − | Z 1 ∩ 2 ∩ Mc x x + 2Mc N (B (x ) B (x ) K) + » »(3Mc + 1) ≤ ε| 1 − 2| εL ε 1 ∩ ε 2 \ ≤ ε

where cε is a constant depending only on %; " and N. Since, for any i = 1;:::;N,

@f t @% ε (x; t) = d… ε (x z)b(z; …) dz ; @x N @x − i Z0 ZIR i the proof of the continuity of @fε=@xi is similar (actually simpler). Let us now set, for any " > 0, u(x) vε(x) = bε(x; t) dt: Z0 f C1 N By applying the chain rule to the composition of the function ε 0 (IR IR) with the BV map x (x; u(x)) (see, for instance, [2, Theorem 3.96]), we get,∈ for any× C1(IRN ), 7→ ∈ 0 u(x) @bε (x)vε(x) dx = (x)dx (x; t) dt (x)bε(x; u(x)) u(x) dx N ∇ − N @x − N ∇ ZIR ZIR Z0 ZIR u+(x) c N 1 (x)bε(x; u(x)) dD u (x)•u(x)d − bε(x; t) dt : − N − H ZIR ZJu Zu−(x) Using again Scorza Dragoni’s theoreme one can easily prove (see, for instance the proof of formula (42) in [18]) that there exists a set Z IRN , with N (Z ) = 0, such that 1 ⊂ L 1 N lim bε(x; t) = b(x; t) for any (x; t) (IR Z1) IR : ε 0+ ∈ \ × → From this equality we immediately get that

lim (x)bε(x; u(x)) u dx = (x)b(x; u(x)) u dx ε 0+ N ∇ N ∇ → ZIR ZIR and that u(x) lim (x)vε(x) dx = (x) dx b(x; t) dt : ε 0+ N ∇ N ∇ → ZIR ZIR Z0 V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV 181 @b Notice that, by assumption (jj), for any t IR the functions ε ( ; t) converge in L1(IRN ) ∈ @x · @b @b to ( ; t) and ε ( ; t) L for any " > 0 and t IR. Therefore, denoting by @x · @x · L1(IRN ) ≤ ∈ I x ; u x u x > u x ; ( ) the interval (0 ( )), if ( ) 0, and the interval ( ( ) 0), otherwise, by Fubini’s theorem and by the Lebesgue’s theorem of dominated convergence, we get

u(x) @b u(x) @b x dx ε x; t dt x dx x; t dt lim sup ( ) @x ( ) ( ) @x( ) ε 0+ IRN 0 − IRN 0 → ¬Z Z Z Z ¬ ¬ @b @b ¬ ¬ dt ¿ t ε x; t x; t dx : ¬ ¬ lim I(x)( ) ( ) ( ) = 0 ¬ ≤ ε 0+ N @x − @x → ZIR ZIR ¬ ¬ ¬ ¬ ¬ ¬ Let us now prove that ¬ ¬

c c lim (x)bε(x; u(x)) dD u = (x)b(x; u(x)) dD u : (10) ε 0 N N → ZIR ZIR To this aim, let us use the coareae formula (8), thus getting e

Dcu c (x)bε(x; u(x) dD u = (x)bε(x; u(x)) (x) d Du (11) N Du IR Cu u | | Z Z \D | | Dcu N 1 e = dt e(x)b (x; u(x))¿ (x) (x) d − ε Cu u Du IR u t u+ \D H Z Z{ −≤ ≤ } | | Dcu dt x b ex; t x d N 1 : = ( ) ε( ) Du ( ) − IR u=t (Cu u) H Z Z{ }∩ \D | | N 1 e N Let us ﬁx t IR. Since − -a.e. x IR is a point of approximate continuity for ∈ H N∈ 1 b( ; t), bε(x; t) converges to b(x; t) for − -a.e. x, hence bε(x; t) converges to b(x; t) for ·N 1 H N − -a.e. x. Moreover, since u BV (IR ), by the coarea formula we have H ∈ e + ∞ N 1 − ( u = t (C )) dt = Du (C ) < : (12) H { } ∩ u \Du | | u \Du ∞ Z−∞ Therefore, for 1-a.e. t IR,e we have L ∈ Dcu Dcu N 1 N 1 lim (x)bε(x; t) (x) d − = (x)b(x; t) (x) d − : ε 0 Du Du u=t (Cu u) H u=t (Cu u) H → Z{ }∩ \D | | Z{ }∩ \D | | From thise equation, passing to the limit in (11) ande using (12) and the dominated convergence theorem, we immediately get (10). Finally, let us consider the Radon measure ¶ = Du 1 and deﬁne for any i; j IN, | | × L ∈ G x; t N b y; t q dy < 1 ; i,j = ( ) IR IR : lim sup ( ) i j ∈ × r 0 −B (x) | − | n → Z r o

where qi is a dense sequence in IR. The set

G = ∞ ∞ G ∩j=1 ∪i=1 i,j 182 V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV

is a Borel set and (x; t) G if and only if x is a point of approximate continuity for ∈ N 1 N b( ; t). By assumption, the function b( ; t) is weakly diﬀerentiable, hence − ( x IR : (x;· t) G ) = 0. Therefore, the complement· Gc of G satisﬁes the equalityH { ∈ 6∈ }

c ¶(G ) = dt ¿Gc (x; t)d Du = 0 : (13) N | | ZIR ZIR Let us estimate

u+(x) u+(x) N 1 N 1 (x)•u(x) d − bε(x; t) dt (x)•u(x) d − b(x; t) dt (14) J H u (x) − J H u (x) ¬Z u Z − Z u Z − ¬ ¬ u+(x) ¬ ¬ ¬ d Du bε(x; t) b(x; t) dt : ≤ k k∞ | | − | − | ZJu Zu−(x) By (13) we have that

u+(x) lim bε(x; t) b(x; t) dt = 0 for Du -a.e. x Ju ε 0 − | − | | | ∈ → Zu−(x) and from this equality we immediately get that the right hand side in (14) goes to zero, as " 0. → This concludes the proof. Remark 3.1. From the above proof it is clear that the integration by parts formula stated in Proposition 1.2 still holds if we replace b by any Borel function ‹ : IRN IR IR such N 1 N × → that, for any t IR, b(x; t) = ‹(x; t) for − -a.e. x IR . ∈ H ∈ The proof of Theorem 1.1 is based on an argument introduced in [15].

Proof of Theorem 1.1. Step 1. We treat separately the two terms depending on the diﬀuse part of Du, i.e. a c j D u + D u, and the jump term D u. Let (un) be a sequence in BV (Ω) converging in L1 u BV – C1 (Ω) to (Ω). Let us ﬁx an open set Ω0 Ω and a function 0 (IR), with 0 –(t) 1.∈ Let K ;K be two compact sets such⊂⊂ that ∈ ≤ ≤ 1 2

K Ω0 C ;K Ω0 C : (15) 1 ⊂ ∩ u 2 ⊂ \ u

Then, we may ﬁnd two open sets Ω1; Ω2, contained in Ω0, such that

Ω Ω = ;K Ω ;K Ω : (16) 1 ∩ 2 ∅ 1 ⊂ 1 2 ⊂ 2

Finally, let us denote by gj the sequence of functions provided by Lemma 2.1. Since

lim inf F (un; Ω) lim inf F (un; Ω1) + lim inf F (un; Ω2) ; (17) n n n →∞ ≥ →∞ →∞ we are going to estimate separately the two terms on the right hand side of this inequality.

Step 2. Let us ﬁx a ﬁnite family Aj j J of disjoint open sets with the closure contained ’ { C}1 ∈ ’ r j J in Ω1. Let ( r)r IN be a sequence in 0 (Ω1), with 0 r 1 for all , and, for any , ∈ ≤ ≤ ∈ V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV 183

– C1 A – j; s let ( j,s)s IN be a sequence in 0 ( j IR), with 0 j,s 1 for all . Since by Lemmas ∈ 2.1 and 2.2 × ≤ ≤ f(x; u; ‚) g (x; u; ‚)– (x; u)’ (x) for all (x; u; ‚) (Ω Z) IR IRN ≥ j j,s r ∈ \ × × j J X∈ f ∞(x; u; ‚) « (x; u);‚ – (x; u)’ (x) for all (x; u; ‚) Ω IR IRN ; ≥ h j i j,s r ∈ × × j J X∈ for any r; s IN, we have, by Proposition 1.2 and Remark 3.1, ∈

F (u ; Ω ) a (x; u )– (x; u )’ dx + a (x; u )– (x; u ); u ’ dx n 1 ≥ 0,j n j,s n r h j n j,s n ∇ ni r j J Ω1 j J ( Ω1 X∈ Z X∈ Z Dcu + « (x; u ); n – (x; u )’ d Dcu h j n Dcu i j,s n r | n| ZΩ1 | n| + un (x) e N 1 e + ’rd − «j(x; t);•u(x) –j,s(x; t)dt Ju Ω H un−(x) h i Z ∩ 1 Z )

un(x) = a (x; u )– (x; u )’ dx dx a (x; t); ’ (x) – (x; t)dt 0,j n j,s n r − h j ∇ r i j,s j J Ω1 j J ( Ω1 0 X∈ Z X∈ Z Z un(x) + ’rdx divx aj(x; t)–j,s(x; t) dt : ZΩ1 Z0 ) ¡ Passing to the limit in this inequality and using Proposition 1.2 and Remark 3.1 again, we easily get

lim inf F (un; Ω1) a0,j(x; u)–j,s(x; u) + aj(x; u)–j,s(x; u); u ’r dx (18) n ≥ h ∇ i →∞ j J ZΩ1 X∈ ¢ £ Dcu + « (x; u(x))– (x; u(x)); ’ d Dcu h j j,s Dcu i r | | j J Ω1 X∈ Z | | eu+(x) e N 1 + «j(x; t)–j,s(x; t);•u(x) dt ’r d − : Ω Ju u (x) h i H j J 1∩ ´ − µ X∈ Z Z ’ C1 ’ x From Lusin’s theorem there exists a sequence r 0 (Ω1), with 0 r( ) 1 such ’ x ¿ x Du x ∈ ≤ ≤r that r( ) Cu Ω ( ) for -a.e. Ω1. Therefore passing to the limit as , ∩ 1 inequality (18)→ becomes | | ∈ → ∞

lim inf F (un; Ω1) a0,j(x; u)–j,s(x; u) + aj(x; u)–j,s(x; u); u dx n ≥ h ∇ i →∞ j J ZΩ1 X∈ ¢ £ Dcu + « (x; u(x))– (x; u(x)); d Dcu : h j j,s Dcu i | | j J Ω1 X∈ Z | | j J – x; t › x e– t ›e x ¿ x Next, taking for any , j,s( ) = j,s( ) ( ), with j,s( ) converging to Dj ( ) + ¿ (x) for Du -a.e. x ∈A , where Cj | | ∈ j D = x A : g (x; u(x); u(x)) > 0 j { ∈ j ∩ Du j ∇ } 184 V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV Dcu C = x A (C ): « (x; u(x)); > 0 ; j ∈ j ∩ u \Du h j Dcu i n | | o we get immediately that e

lim inf F (un; Ω1) –(u) max gj(x; u(x); u(x)); 0 dx n ≥ { ∇ } →∞ j J Aj X∈ Z Dcu + –(u(x)) max « (x; u(x)); ; 0 d Dcu : h j Dcu i | | j J Aj X∈ Z n | | o e e Therefore, by applying Lemma 2.3 with ¶ = Du we obtain, by Lemmas 2.1 and 2.2, | | Dcu lim inf F (u ; Ω ) f(x; u; u)–(u) dx + f ∞ x; u; –(u) d Dcu : n 1 Dcu n ≥ Ω ∇ Ω | | →∞ Z 1 Z 1 ° | |± Step 3. Let us fix a finite family Uj j J of disjoint open sets with compacte closure in ’ { C} 1∈ ’ r – Ω2 IR. Let ( r)r IN be a sequence in 0 (Ω2), with 0 r 1 for all , and let ( j,s)s IN × C∈ 1 U – j; s≤ ≤ ∈ be a sequence in 0 ( j), with 0 j,s 1 for all . Arguing as in Step 2 and letting ’ x ¿ x Du≤ ≤ x r( ) converge to Ju Ω ( ) for -a.e. Ω2, we get by Fubini’s theorem ∩ 2 | | ∈ F u ; « x; t – x; t ;• x ¿ t d ; lim inf ( n Ω2) j( ) j,s( ) u( ) [u (x),u+(x)]( ) n ≥ h i − →∞ j J Ω2 IR X∈ Z × N 1 1 where denotes the product measure of the two »-finite measures − Ju and . Let NH L Am be an increasing sequence of Borel sets such that mAm = IR IR and (Am) < for any m. Let us fix m and, for any j J, let us∪ apply Lusin’s× theorem again∞ to – x; t ∈ – t ¿ x; t S x; t U get a sequence j,s( ) converging -a.e. to ( ) S Am ( ), where j = ( ) j : j ∩ « (x; t);• (x) > 0 . Thus, from the previous inequality we obtain { ∈ h j u i } F u ; – t ¿ t « x; t ;• x ; d lim inf ( n Ω2) ( ) [u (x),u+(x)]( ) max j( ) u( ) 0 n ≥ − {h i } →∞ j J Uj Am X∈ Z ∩ and letting m → ∞ F u ; – t ¿ t « x; t ;• x ; d : lim inf ( n Ω2) ( ) [u (x),u+(x)]( ) max j( ) u( ) 0 n ≥ − {h i } →∞ j J Uj X∈ Z N 1 1 Therefore, by applying Lemma 2.3 with ¶ = = − Ju we obtain, by Lemma 2.2 and Fubini’s theorem, H ×L

u+(x) N 1 lim inf F (un; Ω2) –(t)f ∞(x; t; •u(x))dt d − : n ≥ Ω u (x) H →∞ Z 2 hZ − i Letting –(t) 1 for any t IR, from this inequality and from (19), we obtain, recalling (15), (16) and↑ (17), ∈ lim inf F (un; Ω) F (u; K1) + F (u; K2) : n →∞ ≥ The result follows by letting ﬁrst K C and then K Ω0 C and, ﬁnally, letting 1 ↑ u 2 ↑ \ u Ω0 Ω. ↑ V. De Cicco, N. Fusco, A. Verde / On L1-Lower Semicontinuity in BV 185

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