a remark on the l 1-lower semicontinuity for integral functionals in bv

manuscripta math. 112, 313–323 (2003) © Springer-Verlag 2003

Nicola Fusco · Flavia Giannetti · Anna Verde

A remark on the L1-lower semicontinuity for integralfunctionals in BV

Received: 16 May 2002 / Published online: 15 October 2003

Abstract. We study the L1-lower semicontinuity in BV of an integral functional of the

type∫�

f (x, u,∇u)dx +∫�

−∫ u+(x)

u−(x)f∞(x, t,

Dsu

|Dsu| )dtd|Dsu|. Our assumptions on f

extend previous results recently obtained by Gori, Maggi and Marcellini in the case wherethe above functional is restricted to W 1,1.

1. Introduction

The L1-lower semicontinuity of an integral functional G of the type

G(u,�) =∫�

f (x, u,∇u) dx, (1)

where � is an open subset of RN and u ∈ W 1,1(�), has been extensively studied

in the past years. The starting point of most of the recent studies on this subjectis a celebrated result by Serrin. In [11] he proved that the functional G is lowersemicontinuous in W 1,1(�) with respect to the L1-convergence of u under theassumptions that

f : �× R × RN → [0,+∞) is continuous, (2)

f (x, t, ·) is convex in RN for every (x, t) ∈ �× R, (3)

and that one of the following conditions holds:

(i) f (x, t, ξ) → +∞ as |ξ | → +∞ for every (x, t) ∈ �× R;(ii) f (x, t, ·) is strictly convex for every (x, t) ∈ �× R;

(iii) the derivatives fx , fξ , fxξ exist and are continuous.

After Serrin’s paper, many authors have generalized his result by weakening eitherthe continuity assumption on f or one of the conditions (i)–(iii) above (see forinstance [5], [6], [1], [3], [7], [8], [9], [10], [4]). In particular, two recent papers,one by Gori and Marcellini ([9]), the other one by Gori, Maggi and Marcellini([10]), have shown that condition (iii) can be replaced by a significantly weaker

N. Fusco, F. Giannetti, A. Verde: Dipartimento di Matematica e Applicazioni, Via Cintia,80126 Napoli, Italy.e-mail: {n.fusco,giannett,[email protected]}

DOI: 10.1007/s00229-003-0400-6

314 N. Fusco et al.

assumption. Namely, in ([10]) they prove that the L1-lower semicontinuity of Gin W 1,1 still holds if one replaces (2) with the assumption that f is a nonnegative,locally bounded, Caratheodory function and if (iii) is replaced by the assumptionthat for any open subset �′ whose closure is contained in � and any compactH ⊂ R × R

N , there exists a constant L(�′, H) such that∫�′

∣∣∣∂f∂x(x, t, ξ)

∣∣∣ dx ≤ L(�′, H) for every (t, ξ) ∈ H. (4)

A similar, though more general, condition is also considered in [4]. Moreover, in([9]) and ([10]) it is also proved that their result is sharp in the sense that (4) cannotbe replaced by a weaker condition such as Holder continuity in x, uniform withrespect to (t, ξ).

Here, in the same spirit of the papers [9], [10], [4], we study the lower semi-continuity in L1 of the functional

F(u,�) =∫�

f (x, u,∇u)dx +∫�

f∞(x, u,Dcu

|Dcu| ) d|Dcu|

+∫Ju

[∫ u+(x)

u−(x)f∞(x, t, νu)dt

]dHN−1 , (5)

where u ∈ BV (�), f∞ denotes the recession function of f with respect to ξ ,Dcu is the Cantor part of the measureDu,Dcu/|Dcu| is the derivative of the mea-sure Dcu with respect to its total variation |Dcu| and Ju is the jump set of u (thedefinition of all these quantities are recalled in Section 2).

Namely, we prove the following result.

Theorem 1. Let f be a function satisfying the assumptions (2), (3) and (4). Thenthe functional F defined in (5) is lower semicontinuous in BV (�) with respect tothe L1(�) convergence.

As in the papers [9], [10], [4] quoted above the proof of this result relies on anapproximation lemma (see Lemma 2) due to De Giorgi ([5]) and on the validity

of the chain rule formula for the function x �→∫ u(x)

0b(x, t)dt , with u in BV (see

Lemma 5). However, differently from those papers, our proof is based on the useof suitably chosen test functions which allow us to treat separately the diffuse partof the distributional gradientDu of u (that is the sum of the absolutely continuouspart and of the Cantor part of Du) and the jump part of the gradient. After havingproved separately the lower semicontinuity with respect to the diffuse part and theto jump part, one recovers the whole functional by means of a simple localizationlemma (see Lemma 4).

We observe that the continuity of the integrand f with respect to x is needed inorder to treat the two terms of the functional F containing the singular part of Du

∫�

f∞(x, u(x),Dcu

|Dcu| ) d|Dcu| +

∫Ju

[∫ u+(x)

u−(x)f∞(x, t, νu(x))dt

]dHN−1(x).

However, as in [10], also with our approach the continuity of f with respect to x canbe dropped if we restrict the functional F to the spaceW 1,1(�) (see Proposition 7).

L1-lower semicontinuity in BV 315

2. Preliminary results

We collect in this section a few technical results and some basic material on BVfunctions which will come into play in our proof of Theorem 1. Definitions areclassical, and are recalled mainly to fix notations. We have followed here the nota-tions used in [2]. The reader may use this book as a reference for all the propertiesof BV functions used in the sequel.

The first lemma is a classical approximation result due to De Giorgi, which isstated here in a slightly different version from the original one contained in [5].

Lemma 2. Let f be a continuous function from �× R × RN into [0,+∞), satis-

fying (3). Then there exists a sequence (gj )j∈N of functions from�× R × RN into

R of the type

gj (x, t, ξ) = a0,j (x, t)+N∑i=1

ai,j (x, t)ξi,

with a0,j , ai,j continuous in �× R, such that for all (x, t, ξ) ∈ �× R × RN

f (x, t, ξ) = supj

max{gj (x, t, ξ), 0}.

Moreover, if f satisfies (4), then for any open set�′ ⊂⊂ �, any compact setK ⊂ R

and any j ∈ N there exists Lj (�′,K) such that for any t ∈ K∫�′

∣∣∣∂ai,j∂x

∣∣∣ dx ≤ Lj (�′,K) for any i = 1, . . . , N. (6)

Proof. For the proof of the first part of the statement we refer to the original proofby De Giorgi in [5]. The proof of (6) follows immediately from the fact that in [5]the functions ai,j are defined by setting for all i = 1, . . . , N , j ∈ N,

ai,j (x, t) = −∫

RNf (x, t, ζ )∇iαj (ζ ) dζ,

where αj ∈ C∞0 (R

N) is a suitable mollifier, with αj ≥ 0 and∫RNαj = 1. �

Let us recall that if h : RN → R is a convex function, its recession function

h∞ : RN → R is defined by setting

h∞(ξ) = limt→+∞

h(tξ)

t, for any ξ ∈ R

N. (7)

If f satisfies (3), by f∞(x, t, ξ) we denote, for any (x, t) ∈ � × R the reces-sion function of the convex function f (x, s, ·). The following result is an easyconsequence of the definition (7) and of Lemma 2 (see also [2, Lemma 2.33]).

Lemma 3. Let f be a continuous function from � × R × RN into [0,∞) satis-

fying (3) and let (gj )j∈N be the sequence provided by Lemma 2. Then, for any(x, t, ξ) ∈ �× R × R

N

f∞(x, t, ξ) = supj

max{〈aj (x, t), ξ〉, 0}.

316 N. Fusco et al.

Notice that from this lemma it follows that the function (x, t, ξ) �→ f∞(x, t, ξ)is lower semicontinuous. Thus, the functional F given by (5) is well defined.

The following result is contained in [2, Lemma 2.35].

Lemma 4. Let µ be a positive Radon measure in an open set � ⊂ RN and let

ψj : � → [0,∞], j ∈ N, be Borel functions. Then

∫�

supj

ψj dµ = sup

{∑j∈J

∫Aj

ψj dµ

},

where the supremum ranges among all finite sets J ⊂ N and all families {Aj }j∈N

of pairwise disjoint open sets with compact closure in �.

Let u be a function in L1loc(�). We say that u is approximately continuous at the

point x ∈ � if there exists u(x) ∈ R such that

limr→0

−∫Br(x)

|u(y)− u(x)| dy = 0.

The set Cu of all points where u is approximately continuous is a Borel set. Wesay that a point x ∈ � \ Cu is an approximate jump point for u if there existu+(x), u−(x) ∈ R and νu(x) ∈ SN−1 such that u−(x) < u+(x) and

limr→0

−∫B+r (x;νu(x))

|u(y)− u+(x)|dy=0, limr→0

−∫B−r (x;νu(x))

|u(y)− u−(x)|dy=0 ,

where B+r (x; νu(x)) = {y ∈ Br(x) : 〈y − x, νu(x)〉 > 0} and B−

r (x; νu(x)) isdefined analogously. Also the set Ju ⊂ � \ Cu of all the approximate jump pointsis a Borel set and the function (u+(x), u−(x), νu(x)) : Ju → R × R × SN−1 is aBorel function. Notice that in the sequel we shall write u+(x) or u−(x) also whenx is in Cu, with the obvious meaning that u±(x) = u(x).

Given a point x ∈ Cu, we say that u is approximately differentiable at x if thereexists ∇u(x) ∈ R

N such that

limr→0

1

rN+1

∫Br(x)

|u(y)− u(x)− 〈∇u(x), y − x〉| dy = 0.

The vector ∇u(x) is called the approximate differential of u at x. The set of pointsin Cu where the approximate differential of u exists is a Borel set denoted by Du.It can be easily verified that ∇u : Du → R

N is a Borel map.A function u ∈ L1(�) is called of bounded variation if its distributional gradi-

entDu is an RN -vector valued measure and the total variation |Du| ofDu is finite

in�. The space of all functions of bounded variation in� is denoted by BV (�). Ifu ∈ BV (�), we denote byDau the absolutely continuous part ofDu with respectto the Lebesgue measure LN . The singular partDsu can be split in two more parts,the jump part Dju and the Cantor part Dcu, defined by

Dju = Dsu Ju, Dcu = Dsu−Dju.


Furthermore,

Dau=∇uLN Du, Dcu=Du (Cu\Du),(8)

Dju=(u+−u−)νuHN−1 Ju,

where HN−1 denotes the (N − 1)-dimensional Hausdorff measure in RN (see [2,

Proposition 3.92]).The following lemma provides a useful integration by parts formula. Here and

in the sequel we set for any continuous function h

−∫ b

a

h(t) dt =

1

b − a

∫ b

a

h(t) dt if a < b

h(a) if a = b.

Lemma 5. Let b : RN × R → R be a continuous function with compact support

and let us assume that there exists L such that for any t ∈ R∫RN

∣∣∣ ∂b∂x(x, t)

∣∣∣ ≤ L. (9)

Then, for any u ∈ BV (RN) and any ϕ ∈ C10(R

N), we have

−∫

RN

(∫ u(x)

0b(x, t)dt

)∇ϕ dx =

∫RNb(x, u)ϕ∇u dx

+∫

RN

[−∫ u+(x)

u−(x)b(x, t)dt

]ϕdDsu(x)+

∫RNϕdx

∫ u(x)

0

∂b

∂x(x, t) dt. (10)

Proof. If∂b

∂xis continuous, (5) is an immediate consequence of the chain rule

formula given in [2, Theorem 3.96] applied to the composition of the functionf : R

N × R → R, defined by

f (x, t) =∫ t

0b(x, τ ) dτ for all (x, t) ∈ R

N × R,

with the function x ∈ RN �→ (x, u(x)). The general case follows by applying (10),

with b replaced by

bε(x, t) =∫

RN�(y)b(x + εy, t) dy for all (x, t) ∈ R

N × R,

where � is a standard mollifier, and then passing to the limit as ε → 0. In fact, sinceb is continuous, bε(x, t) → b(x, t), as ε → 0+, for all (x, t) ∈ R

N ×R. Thereforeby applying (10) with b replaced by bε, the integral on the left hand side and thetwo first integrals on the right hand side converge to the corresponding integrals

for b. Since∂bε

∂x(x, t) converge to

∂b

∂x(x, t) in L1(RN) for any t ∈ R, we get that

limε→0+

∫RN

∣∣∣∂bε∂x(x, t)− ∂b

∂x(x, t)

∣∣∣ dx = 0 for all t ∈ R. (11)

318 N. Fusco et al.

Moreover, since by (9) the functions t �→∫

RN

∣∣∣∂bε∂x(x, t)− ∂b

∂x(x, t)

∣∣∣dx are

bounded from above by 2L uniformly with respect to ε and vanish outside a compactsubset of R, we have from (11)

limε→0+

∫R

dt

∫RN

∣∣∣∂bε∂x(x, t)− ∂b

∂x(x, t)

∣∣∣ dx = 0.

From this equation, using Fubini’s theorem we easily get that for any ϕ ∈ C10(R

N)

limε→0+

∫RNϕ dx

∫ u(x)

0

∂bε

∂x(x, t) dt =

∫RNϕ dx

∫ u(x)

0

∂b

∂x(x, t) dt. (12)

Hence, the result follows. �

Remark 6. It is clear that the above lemma still holds if we assume thatb is a boundedCaratheodory function in R

N × R with compact support and if u ∈ W 1,1(RN).In fact (12) can be proved exactly as before and, in this case, the same argumentleading to the proof of (12) gives also that

limε→0+

∫RN

(∫ u(x)

0bε(x, t)dt

)∇ϕ dx =

∫RN

(∫ u(x)

0b(x, t)dt

)∇ϕ dx.

Moreover, it is not difficult to prove (see the proof of Lemma 8 in [10]) that thereexists a measurable set C ⊂ R

N such that LN(RN \ C) = 0 and

limε→0+

bε(x, t) = b(x, t) for any (x, t) ∈ C × R.

From this equation one gets immediately, since b is bounded,

limε→0+

∫RNbε(x, u)ϕ∇u dx =

∫RNb(x, u)ϕ∇u dx ,

thus proving (10).

3. Proof of Theorem 1.1

In this section we are going to prove the main result of the paper, Theorem 1. Ourproof is based on the approximation Lemma 2, and uses the integration by partsformula (10) as well as Lemma 4. For the general case of a functional defined onBV we have to treat separately the two terms depending on the diffuse part ofDu,i.e. Dau + Dcu, and the jump term. However the reader may check that in thespecial case of a functional of the type (1), the proof below becomes simpler thanother proofs known in the literature (see e.g. [9], [10]).


Proof of Theorem 1. Step 1. Let (un)be a sequence inBV (�) converging inL1(�)

to u ∈ BV (�). In order to prove that the functional F defined by (5) is lower semi-continuous along the sequence (un) we may assume, without loss of generality,that un(x) → u(x) for LN -a.e. x ∈ �. Let us fix also an open set �′ ⊂⊂ � and afunction η ∈ C1

0(R), with 0 ≤ η(t) ≤ 1. LetK1,K2 be two compact sets such that

K1 ⊂ �′ ∩ Cu, K2 ⊂ �′ \ Cu. (13)

Then, we may find two open sets �1, �2, contained in �′, such that

�1 ∩�2 = ∅, K1 ⊂ �1, K2 ⊂ �2. (14)

Finally, let us denote by gj the sequence of functions provided by Lemma 2. Since

lim infn→∞ F(un,�) ≥ lim inf

n→∞ F(un,�1)+ lim infn→∞ F(un,�2), (15)

we are going to estimate separately the two terms on the right hand side of thisinequality.

Step 2. Let us fix a finite family {Aj }j∈J of disjoint open sets with the closurecontained in �1. Let (ϕr)r∈N be a sequence in C1

0(�1), with 0 ≤ ϕr ≤ 1 for all r ,and, for any j ∈ J , let (ηj,s)s∈N be a sequence in C1

0(Aj × R), with 0 ≤ ηj,s ≤ 1for all j, s. Since f (x, t, ξ) ≥ ∑

j∈J gj (x, t, ξ)ηj,s(x, t)ϕr(x) and, by Lemma 3,f∞(x, t, ξ) ≥ ∑

j∈J 〈aj (x, t), ξ〉ηj,s(x, t)ϕr(x) for any r, s ∈ N, we have

lim infn→∞ F(un,�1) ≥

∑j∈J

lim infn→∞

∫�1

a0,j (x, un)ηj,s(x, un)ϕr dx

+∑j∈J

lim infn→∞

{∫�1

〈aj (x, un)ηj,s(x, un),∇un〉ϕrdx

+N∑i=1

∫�1

[−∫ u+

n (x)

u−n (x)

ai,j (x, t)ηj,s(x, t)dt

]ϕrdD

si u

}. (16)

Since un(x) → u(x) for LN -a.e. x ∈ �, we have for any j ∈ J, r, s ∈ N

limn→∞

∫�1

a0,j (x, un)ηj,s(x, un)ϕr dx =∫�1

a0,j (x, u)ηj,s(x, u)ϕr dx. (17)

Since the functions ai,j (x, t)ηj,s(x, t) satisfy the assumptions of Lemma 5, for alli = 1, . . . , N , j ∈ J , r ∈ N, using again the LN -a.e. convergence ofun(x) → u(x)

and the fact that

∫�×R

|divx(aj (x, t)ηj,s(x, t))| dxdt < ∞,

320 N. Fusco et al.

we have

limn→∞

{∫�1

〈aj (x, un)ηj,s(x, un),∇un〉ϕrdx

+N∑i=1

∫�1

[−∫ u+

n (x)

u−n (x)

ai,j (x, t)ηj,s(x, t)dt

]ϕrdD

si u

}

= limn→∞

{−

∫�1

〈∫ un(x)

0aj (x, t)ηj,s(x, t)dt,∇ϕr 〉 dx

−∫�

ϕrdx

∫ un(x)

0divx(aj (x, t)ηj,s(x, t)) dt

}

= −∫�1

〈∫ u(x)

0aj (x, t)ηj,s(x, t)dt,∇ϕr 〉 dx

−∫�

ϕrdx

∫ u(x)

0divx(aj (x, t)ηj,s(x, t)) dt

=∫�1

〈aj (x, u)ηj,s(x, u),∇u〉ϕrdx

+N∑i=1

∫�1

[−∫ u+(x)

u−(x)ai,j (x, t)ηj,s(x, t)dt

]ϕrdD

si u,

for any j, r, s. Therefore, from this equation and from (16), (17), we get, splittingDsu as the sum of its Cantor part and jump part,

lim infn→∞ F(un,�1)

≥∑j∈J

∫�1

[a0,j (x, u)ηj,s(x, u)+ 〈aj (x, u)ηj,s(x, u),∇u〉

]ϕr dx

+∑j∈J

∫�1

〈aj (x, u(x))ηj,s(x, u(x)), Dcu

|Dcu| 〉ϕr d|Dcu|

+∑j∈J

∫�1∩Ju

[∫ u+(x)

u−(x)〈aj (x, t)ηj,s(x, t), νu(x)〉dt

]ϕr dHN−1. (18)

From Lusin’s theorem it follows that there exists a sequence ϕr ∈ C10(�1), with

0 ≤ ϕr(x) ≤ 1 such that ϕr(x) → χCu∩�1(x) for |Du|-a.e. x ∈ �1. With such achoice of ϕr , passing to the limit as r → ∞ in (18), we get


∑j∈J

∫�1


]dx

+∑j∈J

∫�1


|Dcu| 〉 d|Dcu|.


From this inequality, taking, for any j ∈ J , ηj,s(x, t) = γj,s(x)η(t), with γj,s(x)converging to χDj (x)+ χCj (x) for |Du|-a.e. x ∈ Aj , where

Dj = {x ∈ Aj ∩ Du : gj (x, u(x),∇u(x)) > 0}Cj =

{x ∈ Aj ∩ (Cu \ Du) : 〈aj (x, u(x)), D

cu

|Dcu| 〉 > 0},

we get immediately that


∑j∈J

∫Aj

η(u)max{gj (x, u(x),∇u(x)), 0} dx

+∑j∈J

∫Aj

η(u(x))max{〈aj (x, u(x)), D

cu

|Dcu| 〉, 0}d|Dcu|.

Therefore, by applying Lemma 4 with µ = |Du| and

ψj (x) = η(u(x))max{gj (x, u(x),∇u(x)), 0}χDu(x)

+η(u(x))max{〈aj (x, u(x)), D

cu

|Dcu| 〉, 0}χCu\Du

(x),

we obtain, by Lemma 2 and Lemma 3,


∫�1

f (x, u,∇u)η(u) dx

+∫�1

f∞(x, u,

Dcu

|Dcu|)η(u) d|Dcu|. (19)

Step 3. Let us fix a finite family {Uj }j∈J of disjoint open sets with compact closurein�2 ×R. Let (ϕr)r∈N be a sequence in C1

0(�2), with 0 ≤ ϕr ≤ 1 for all r , and let(ηj,s)s∈N be a sequence in C1

0(Uj ), with 0 ≤ ηj,s ≤ 1 for all j, s. Arguing exactlyas in Step 2, we obtain, in place of (18),


≥∑j∈J

∫�2


]ϕr dx

+∑j∈J

∫�2


|Dcu| 〉ϕr d|Dcu|

+∑j∈J

∫�2∩Ju

[∫ u+(x)

u−(x)〈aj (x, t)ηj,s(x, t), νu(x)〉dt

]ϕr dHN−1.

Letting ϕr(x) converge to χJu∩�2(x) for |Du|-a.e. x ∈ �2, we get by Fubini’stheorem


∑j∈J

∫�2×R

〈aj (x, t)ηj,s(x, t), νu(x)〉χ[u−(x),u+(x)](t) dλ,

322 N. Fusco et al.

where λ denotes the product measure of the two σ -finite measures HN−1 Ju andL1. LetAm be an increasing sequence of Borel sets such that ∪mAm = R

N ×R andλ(Am) < ∞ for any m. Let us fix m and, for any j ∈ J , let us apply Lusin’s theo-rem again to get a sequence ηj,s(x, t) converging λ-a.e. to η(t)χSj∩Am(x, t), whereSj = {(x, t) ∈ Uj : 〈aj (x, t), νu(x)〉 > 0}. Thus, from the previous inequality weobtain


≥∑j∈J

∫Uj∩Am

η(t)χ[u−(x),u+(x)](t)max{〈aj (x, t), νu(x)〉, 0} dλ

and letting m → ∞


∑j∈J

∫Uj

η(t)χ[u−(x),u+(x)](t)max{〈aj (x, t), νu(x)〉, 0} dλ.

Therefore, by applying Lemma 4 with µ = λ = HN−1 Ju × L1 and

ψj (x) = η(t)χ[u−(x),u+(x)](t)max{〈aj (x, t), νu(x)〉, 0},

we obtain, by Lemma 3 and Fubini’s theorem,


∫�2×R

η(t)χ[u−(x),u+(x)](t)f∞(x, t, νu(x)) dλ

=∫�2

[∫ u+(x)

u−(x)η(t)f∞(x, t, νu(x))dt

]dHN−1.

Letting η(t) ↑ 1 for any t ∈ R, from this inequality and from (19), we obtain,recalling (13), (14) and (15),

lim infn→∞ F(un,�) ≥ F(u,K1)+ F(u,K2).

The result follows by letting first K1 ↑ Cu and then K2 ↑ �′ \ Cu and, finally,letting �′ ↑ �. �

Proposition 7. Let f : � × R × RN → [0,∞) be a Caratheodory function sat-

isfying the assumptions (3) and (4). Then the functional G defined in (1) is lowersemicontinuous in W 1,1(�) with respect to the L1(�) convergence.

Proof. Notice that if f is a Caratheodory function in �× R × RN then Lemma 2

holds with a0,j , ai,j Caratheodory functions in � × R. It is then clear that ifun, u ∈ W 1,1(�) and un → u in L1(�), then the same argument used in Step1 and Step 2 of the proof of Theorem 1 still works, using Remark 6 instead ofLemma 5. �


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a remark on the l 1-lower semicontinuity for integral functionals in bv

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