Journal of Convex Analysis Volume 1 (1994), No. 2, 151–172

On Lower Semicontinuity in BV Setting

Primo Brandi, Anna Salvadori

Department of Mathematics, University of Perugia, Via L. Vanvitelli, I-06123 Perugia, Italy. e-mail: mateas@ipguniv.unipg.it

Received 25 January 1994 Revised manuscript received 23 September 1994

Dedicated to the memory of Professor Lamberto Cesari with deepest affection and gratitude

1. Introduction

We discuss here the lower semicontinuity for integral functionals of the type

I[u, v] = ∫_{G} F (x, u(x), v(x)) dx  (1.1)

with respect to a sequence (u_{k}, v_{k})_{k≥0} subjected to the constraints

(x, u_{k}(x)) ∈ A, v_{k}(x) ∈ Q(x, u_{k}(x)) a.e. in G, k ∈ IN  (1.2)

where G ⊂ IR^{ν} is a bounded open set and Q : A × IR^{m} → 2^{IR^{N}} , A ⊂ IR^{ν+n}, is a given multifunction. A great deal of research was devoted to this subject for the topology of L_{p}−convergence of (u_{k})_{k≥0} and weak L_{q}−convergence of (v_{k})_{k≥0}. We only mention Cesari [9e] and for the free case (i.e. Q(t, x) = IR^{N}, (t, x) ∈ A) we refer to [2,5,7,8,9e,12,13,15,16,17,18,23] where also a list of references can be found. We are interested here in a result which involves a weaker topology; more precisely, it should not require any additional convergence assumption on the differential elements of the highest order when applied to the functionals of the calculus of variations. The reason is that this kind of theorems fit in optimization problems where BV (not necessarily continuous) solutions are expected, since the compactness results on BV do not involve the weak convergence of the gradients, as it occurs in Sobolev’s spaces. To this purpose we introduce here the mean value (mv) condition (see Section 2). Roughly speaking a sequence (v_{k})_{k≥0} of summable functions satisfies (mv) provided for a.e. t_{0} ∈ G

lim_{h→0} lim_{k→+∞} − ∫_{B(t_{0},h)}[v_{k}(t) − v_{0}(t)] dt = 0.

ISSN 0944-6532 / $ 2.50 ◯^{c} Heldermann Verlag

152 P. Brandi, A. Salvadori / On lower semicontinuity in BV setting

It is easy to see that (mv) is weaker than L_{1}−convergence, but the main point is that (mv) is the proper assumption for our setting. In fact (see Proposition 3.7) if (x_{k})_{k∈IN} is a sequence in W^{1,1}(G, IR^{n}) which L_{1}–converges to a BV function x_{0}, then we get that a subsequence of the gradients (Dx_{k})_{k≥0} satisfies (mv). Here Dx denotes the “essential gradient” of the BV function x, i.e. the density of the absolutely continuous part of the distributional derivative with respect to the Lebesgue measure. The lower semicontinuity results we present here are obtained as a consequence of a closure theorem for an orientor field of type (1.2) with respect to the convergence

u_{k} L_{1}–converges to u_{0} and (v_{k})_{k∈IN} satisfies (mv).  (1.3)

Our main lower semicontinuity theorem is the following (see Theorem 5.1)

Main result. Assume that A is closed and F is non negative. Let (u_{k}, v_{k})_{k≥0} be a sequence of summable functions such that

i) (t, u_{k}(t)) ∈ A, v_{k}(t) ∈ Q(t, u_{k}(t)) a.e. in G, k ∈ IN;

ii) u_{k} L_{1}–converges to u_{0}  and  (v_{k})_{k∈IN} satisfies (mv).

iii) Suppose that the multifunction

Q̃_{0}(t, x) = {(y^{0}, y) : y_{0} ≥ F (t, x, y), y ∈ Q(t, x)}

satisfies property (Q) and (wF̃) at the point (t_{0}, u_{0}(t_{0})), for a.e. t_{0} ∈ G.

Then (t, u_{0}(t)) ∈ A, v_{0}(t) ∈ Q(t, u_{0}(t)) a.e. in G and

lim inf_{k→+∞} I[u_{k}, v_{k}] ≥ I[u_{0}, v_{0}].

Property (Q), introduced by Cesari in 1966 (see [9e]), is an intermediate condition between upper semicontinuity and Kuratowski property. As we recall in Section 6, it is a seminormality assumption on the integrand and hence implies that F (t, x, ·) is convex.

Condition (wF̃) acts on the second variable, it was introduced in [11c] as a variant of Cesari’s Lipschitz condition (F). Actually, we prove here (see Section 6) that (wF̃) is really a weakening of assumption (F). Some noteworthy particular cases of our functionals are the following

I[x] = ∫_{G} F (t, (Ux)(t), (Lx)(t)) dt  (1.4)

where U and L are given operators not necessarily linear. For example

Lx = Dx, or Lx = div x or Lx = D[Ψ(·, x(·))]

with Ψ a Lipschitzian function. For the details and further examples we refer to Section 7.

We wish to recall that the lower semicontinuity for functionals of type (1.4), with respect to the weak topology in Sobolev’s spaces, was studied by E.Rothe [23], G.Fichera [18], Cesari [9c,d] and Cesari-D.E.Cowles [12]. A particular class of integrals to which our theory apply, is the following

I[u] = ∫_{G} |< a(t), D[Ψ(t, u(t))] > +Φ(t, u(t))| dt

where a is continuous, Ψ is Lipschitzian and Φ is a Carathéodory function such that |Φ(t, x)| ≤ ϕ(t), with ϕ summable. They take their source from problems of conservation laws. In order to prove existence results for the relaxed functional on BV, Cesari [5d] adopted a suitable tranformation which allows to reduce the functional I to standard integrals of the calculus of variations. Now, in force of the present formulation, where abstract operators are involved, we can deal with functional I directly with remarkable advantages both in the assumptions and in the proofs. Note that also functionals of the type

I[u] = ∫_{G} [< a(t), D[Ψ(t, u(t))] > +Φ(t, u(t))]^{+} dt

can be handled in the same way. Finally, we wish to mention that our research is partially motivated by a study of a variational model for the plastic deformation of beams and plates, where functionals of type (1.4) and BV solutions are involved ([11e,14,3a,b,4].)

2. Preliminaries

Let ν, m and p be given integers. Let G ⊂ IR^{ν} be a bounded open set. According to standard notations, we denote by L_{1}(G, IR^{m}) the space of summable functions x : G → IR^{m}, by W^{p,1}(G, IR^{m}) the Sobolev space of the functions x ∈ L_{1}(G, IR^{m}) whose distributional derivatives up to the order p are summable functions and by BV (G, IR^{m}) the space of the functions x ∈ L_{1}(G, IR^{m}) which are of bounded variation in the sense of Cesari [9a].

For m = 1 we will briefly write L_{1}, W^{p,1} and BV, respectively.

A BV function x admits an “essential gradient” [26], i.e. has a.e. partial derivatives \frac{∂x^{i}}{∂t_{j}} computed by usual incremental quotients disregarding the values taken by x on a suitable null set. Moreover the “essential gradient” coincides with the density of the absolutely continuous part of the distributional derivative with respect to the Lebesgue measure. We

denote by Dx = (\frac{∂x^{i}}{∂t_{j}} , i = 1, ..., m, j = 1, ..., ν) and call Dx the gradient of x.

Given a point t_{0} ∈ G and a constant h > 0, we put

q_{h}(t_{0}) = [t_{0} − h, t_{0} + h] = {t ∈ IR^{ν} : t_{0}_{j} − h ≤ t_{0}_{j} ≤ t_{0}_{j} + h, j = 1, ..., ν}

in the case the point t_{0} is clearly determined, we briefly write q_{h}(t_{0}) = q_{h}. Moreover, we denote by |q_{h}| the area of the interval. Given a function x ∈ L_{1}(G, IR^{m}), for every t_{0} ∈ G and h > 0 sufficiently small, we put

− ∫_{q_{h}} x(t) dt = |q_{h}|^{−1} ∫_{q_{h}} x(t) dt.

Given a function x^{0} : R_{0} → IR, where R_{0} is a closed interval, for every subinterval R = [a, b] = {t ∈ IR^{ν} : a_{j} ≤ t_{j} ≤ b_{j}, j = 1, ..., ν}, we consider the difference of order ν relative to the 2^{ν} vertices of R, say

Δ_{R}x^{0} = x^{0}(b) − x^{0}(a) if ν = 1
Δ_{R}x^{0} = x^{0}(b_{1}, b_{2}) − x^{0}(b_{1}, a_{2}) − x^{0}(a_{1}, b_{2}) + x^{0}(a_{1}, a_{2}) if ν = 2

and so on. The function x^{0} is said to be of bounded variation in the sense of Vitali (VBV) [25] provided the interval function Δ_{R}x^{0} has bounded variation. A VBV function has a.e. superficial derivatives, say D^{∗}x^{0}(t_{0}) = lim_{h→0}(2h)^{−ν}Δ_{q_{h}}x^{0}(t_{0}) and D^{∗}x^{0} is a summable

function.

The function x^{0} is said to be absolutely continuous in the sense of Vitali (VAC) [25] if the interval function Δ_{R}x^{0} is absolutely continuous; in this case we have Δ_{R}x^{0} = ∫_{R} D^{∗}x^{0}(t) dt.

3. The mean-value condition

We introduce the following definition for a sequence (v_{k})_{k≥0} in L_{1}(G, IR^{m}).

Definition 3.1. We say that (v_{k})_{k≥0} satisfies the mean value (mv) condition at a point t_{0} ∈ G provided

(mv) there exists a null set H = H(t_{0}) ⊂ IR^{+} such that, for every number ε > 0 a constant 0 < h_{0} = h_{0}(t_{0}, ε) can be determined in such a way that, for every h ∈]0, h_{0}[−H, an integer k_{0} = k_{0}(t_{0}, ε, h) exists such that for every k ≥ k_{0}

| − ∫_{q_{h}} v_{k}(t) dt − v_{0}(t_{0}) | < ε.

We say that (v_{k})_{k≥0} satisfies (mv) condition on G if (mv) holds at a.e. point t_{0} ∈ G.

Let us observe that (mv) can be written:

(mv) there exists a null set H = H(t_{0}) ⊂ IR^{+} such that for i = 1, ..., m

lim_{h→0,h∉H} lim inf_{k→∞} − ∫_{q_{h}} v_{k}^{i}(t)dt = lim_{h→0,h∉H} lim sup_{k→∞} − ∫_{q_{h}} v_{k}^{i}(t)dt = v_{0}^{i}(t_{0}).

P. Brandi, A. Salvadori / On lower semicontinuity in BV setting 155

The following criterions for (mv) condition can be easily proved. Proposition 3.2. Let (v_{k})_{k≥0} and (w_{k})_{k≥0} be two sequences which satisfy (mv) at t_{0} ∈ G, then for every α, β ∈ IR the sequence (αv_{k} + βw_{k})_{k≥0} satisfies (mv) at the same point. Proposition 3.3. Let (v_{k})_{k≥0} be a sequence in L_{1}(G, IR^{m}) and assume that

i) for a.e. t_{0} ∈ G there exists a constant h_{0} = h_{0}(t_{0}) > 0 such that for a.e. h ∈]0, h_{0}[

lim_{k→+∞} ∫_{q_{h}} v_{k}(t) dt = ∫_{q_{h}} v_{0}(t) dt.

Then (v_{k})_{k≥0} satisfies (mv) on G.

As an immediate consequence of Proposition 3.3 we also have Corollary 3.4. If v_{k} ⇀ v_{0} weakly in L_{1}(G, IR^{m}), then the sequence (v_{k})_{k≥0} satisfies (mv) on G. Remark 3.5. On the converse, note that (mv) condition does not imply condition i) in Proposition 3.3 and, a fortiori, weak convergence in L_{1}(G, IR^{m}). To this purpose, let us consider a sequence (u_{k})_{k∈IN} in W^{1,1}(G, IR^{m}) which L_{1}–converges to a function u_{0} ∈ BV (G, IR^{m}) and let v_{k} = Du_{k}, k ≥ 0, be the sequence of the gradients. Of course (Du_{k})_{k≥0} does not satisfy condition i) since, otherwise the limit function would still belong to W^{1,1}(G, IR^{m}). But, by virtue of Proposition 3.7 below, there exists a subsequence of the gradients which satisfies (mv) on G. Still as a consequence of Proposition 3.3 we have Corollary 3.6. Let (u_{k})_{k≥0} be a sequence which converges in L_{1}(G, IR^{m}) and let Φ : A → IR be a Carathéodory function such that

|Φ(t, u_{k}(t))| ≤ ϕ(t), a.e. in G, k ∈ IN

with ϕ ∈ L_{1} . Then the sequence ( Φ(·, u_{k}(·)) )_{k≥0} satisfies (mv) G. Moreover, note that Lemma 2 and 6 in [11d] can be written in terms of (mv) condition as it follows. Proposition 3.7. Let (u_{k})_{k∈IN} be a sequence in W^{1,1}(G, IR^{n}) which L_{1}–converges to a function u_{0} ∈ BV (G, IR^{n}). Then there exists a subsequence of the gradients (Du_{s_{k}})_{k≥0} which satisfies (mv) on G.

Proposition 3.8. Let (u_{k}^{0})_{k∈IN} be a sequence of VAC functions which converges pointwise a.e. in R_{0} to a VBV function u_{0}. Then the superficial derivatives (D^{∗}u_{k})_{k≥0} satisfies (mv) on R_{0}.

Remark 3.9. Note that for ν = 1 and R_{0} = [a, b] the concepts of VAC and VBV reduce to W^{1,1} and BV respectively. Thus, by virtue of Proposition 3.8 the result of Proposition 3.7 improves as follows:

if u_{k} : [a, b] → IR^{m}, k ∈ IN, is a sequence of curves in W^{1,1} which converges pointwise a.e. to a curve u_{0} ∈ BV, then the derivatives (u′_{k})_{k≥0} satisfies (mv) on [a, b].

As an application of Proposition 3.7 we can also prove the following criterion.

Proposition 3.10. Let (u_{k})_{k∈IN} be a sequence in W^{1,1}(G, IR^{n}) which L_{1}–converges to a function u_{0} ∈ BV (G, IR^{n}).

Assume that A ⊂ IR^{ν+n} is a set whose projection in the t-space IR^{ν} contains G and Ψ : A → IR^{m} is a Lipschitzian function. Then there exists a subsequence of the gradients

( D[Ψ(·, u_{s_{k}}(·))] )_{k≥0}

which satisfies (mv) on G.

Proof. Note that the sequence Ψ_{k} : G → IR^{m} defined by Ψ_{k}(t) = Ψ(t, u_{k}(t)), k ≥ 0, satisfies the assumptions of Proposition 3.7.

4. A closure result

Let A ⊂ IR^{ν+n}, with n ∈ IN, be a set whose projection in the t-space IR^{ν} contains G.

Let Q : A → 2^{IR^{m}} be a multifunction with nonempty values and let us consider the multivalued equation

(t, u(t)) ∈ A,  v(t) ∈ Q(t, u(t)),  a.e. in G.  (4.1)

We denote by S_{Q} the set of the measurable solutions of (4.1) and also use the notation S_{Q}^{1} = S_{Q} ∩ L_{1}(G, IR^{n+m}).

We recall that multifunction Q is said to satisfy Cesari’s property (Q) at a point (t_{0}, x_{0}) ∈ A, provided [9e]

Q(t_{0}, x_{0}) = ∩_{σ>0} cl co {∪ Q(t, x), |t − t_{0}| ≤ σ, |x − x_{0}| ≤ σ}.  (Q)

Note that if (Q) holds, then the set Q(t_{0}, x_{0}) is necessarely closed and convex.

Moreover, in [11c] (see also [11d]) the following condition on multifunction Q was introduced.

Let S = (u_{k}, v_{k})_{k∈IN} be a given sequence in S_{Q}^{1}, we say that Q satisfies condition (wF) at a point (t_{0}, x_{0}) ∈ A, with respect to the sequence S, provided (wF) given any number ε > 0, there exist two numbers 0 < σ = σ(t_{0}, x_{0}, S, ε) ≤ ε

and 0 < h_{0} = h_{0}(t_{0}, x_{0}, S, ε) ≤ σ such that for a.e. 0 < h < h_{0} there exist a subsequence (s_{k})_{k∈IN} and a sequence (\overline{u}_{s_{k}}, w_{s_{k}})_{k∈IN} in S_{Q} with the property that for every k ∈ IN

|\overline{u}_{s_{k}}(t) − x_{0}| ≤ σ a.e. in q_{h} and  | − ∫_{q_{h}}[w_{s_{k}}(t) − v_{s_{k}}(t)] dt| ≤ ε.

Besides equation (4.1), also the following type of multivalued equations are involved in problems of the calculus of variations (see [9e])

(t, u(t)) ∈ A, (v^{0}(t), v(t)) ∈ Q̃(t, u(t)),  a.e. in G.  (4.˜1)

where Q̃ : A → 2^{IR^{m+1}} has the following property:

if (y^{0}, y) ∈ Q̃(t, x) and y′ > y^{0}, then (y′, y) ∈ Q̃(t, x).

Let S_{Q} denote the set of the solutions of (4.˜1) and again we put S1 ; _{Q} = S_{Q} ∩ L_{1}(G,

IR^{n+1+m}).

Note that condition (wF) modifies as follows (see [11c,d]).

Given a sequence S = (u_{k}, v_{k}^{0}, v_{k})_{k∈IN} in S1 ; _{Q}, we shall say that Q̃ satisfies condition

(wF̃) at a point (t_{0}, x_{0}) ∈ A, with respect to the sequence S, provided

(wF̃) given any number ε > 0, there exist two numbers 0 < σ = σ(t_{0}, x_{0}, S, ε) ≤ ε and 0 < h_{0} = h_{0}(t_{0}, x_{0}, S, ε) ≤ σ such that for a.e. 0 < h < h_{0} there exist a subsequence (s_{k})_{k∈IN} and a sequence (\overline{u}_{s_{k}}, w_{s_{k}}^{0}, w_{s_{k}})_{k∈IN} in S_{Q} with the property that for every k ∈ IN

|\overline{u}_{s_{k}}(t) − x_{0}| ≤ σ a.e. in q_{h} and
| − ∫_{q_{h}}[w_{s_{k}}(t) − v_{s_{k}}(t)] dt| ≤ ε − ∫_{q_{h}}[w_{s_{k}}^{0}(t) − v_{s_{k}}^{0}(t)] dt ≤ ε.

We are ready to state the following closure result which can be proved by the same technique adopted for Theorem 3 in [11d].

Theorem 4.1. (A closure result). Assume that A is closed.

Let Q̃ : A → 2^{IR^{m+1}} be a multifunction with nonempty values and let (u_{k}, v_{k}^{0}, v_{k})_{k≥0} be a given sequence. Suppose that

i) (u_{k}, v_{k}^{0}, v_{k}) ∈ S1 ; _{Q}, k ∈ IN;

ii) u_{k} L_{1} –converges to u_{0}; iii) (v_{k}^{0}, v_{k})_{k≥0} satisfies (mv) on G;

iv) for a.e. t_{0} ∈ G, multifunction Q̃ has properties (Q) and (wF̃), with respect to the sequence (u_{k}, v_{k}^{0}, v_{k})_{k∈IN}, at the point (t_{0}, u_{0}(t_{0}))

v) if (y^{0}, y) ∈ Q̃(t, x) and y′ > y^{0}, then (y′, y) ∈ Q̃(t, x).

Then the limit function (u_{0}, v_{0}^{0}, v_{0}) belongs to S1 ; _{Q}.

Remark 4.2. Theorem 4.1 also holds in the case A = G × A_{0} with A_{0} ⊂ IR^{n} closed set.

Note that, in the particular case that A = G × A_{0} and multifunction Q̃ : G → 2^{IR^{m+1}} depends only on variable t, assumption (wF̃) is trivially satisfied.

Remark 4.3. As a particular case of Theorem 4.1 a closure result can be proved for equation (4.1), where assumption v) is obviously omitted and condition (wF̃) is replaced by (wF). Again, in the particular case that A = G×A_{0} and multifunction Q : G → IR^{m} depends only on variable t, assumption (wF) is trivially satisfied. Moreover, (wF) holds in the case Q(t, x) = IR^{n}, (t, x) ∈ A = G × A_{0}.

For these and other conditions assuring assumptions (wF) and (wF̃) we refer to [11c,d] and Section 6.

5. Application to the lower semicontinuity of integral functionals

Let M = {(t, x, y) : y ∈ Q(t, x), (t, x) ∈ A} denote the graph of multifunction Q.

We consider a function F : M → IR such that, for every (u, v) ∈ S_{Q}^{1}, F (·, u(·), v(·)) is measurable and F^{−}(·, u(·), v(·)) ∈ L_{1}.

Let I : S_{Q}^{1} ⟶ IR ∪(+∞) be the functional defined by

I(u, v) = ∫_{G} F (t, u(t), v(t)) dt if F^{+}(·, u(·), v(·)) ∈ L_{1}
I(u, v) = +∞ elsewhere.

As an application of closure Theorem 4.1 we shall prove here some lower semicontinuity theorems for the functional I.

To this end, let us consider the multifunction Q̃_{0} : A → 2^{IR^{m+1}} defined by

Q̃_{0}(t, x) = {(y^{0}, y) : y_{0} ≥ F (t, x, y), y ∈ Q(t, x)}.  (5.1)

P. Brandi, A. Salvadori / On lower semicontinuity in BV setting 159

Theorem 5.1. (A lower semicontinuity result). Assume that A is closed. Let (u_{k}, v_{k})_{k∈IN} be a sequence in S_{Q}^{1} and let (u_{0}, v_{0}) be a function in L_{1}(G, IR^{n+m}) such that

i) u_{k} L_{1} –converges to u_{0}; ii) (v_{k})_{k≥0} satisfies (mv) on G. Suppose that iii) a function λ ∈ L_{1} exists such that F_{k}(t) = F (t, u_{k}(t), v_{k}(t)) ≥ λ(t), a.e. in

G, k ≥ 0; iv) for a.e. t_{0} ∈ G multifunction Q̃_{0} has properties (Q) and (wF̃), with respect to

the sequence (u_{k}, F_{k}, v_{k})_{k∈IN}, at the point (t_{0}, u(t_{0})). Then the couple (u_{0}, v_{0}) lies in S_{Q}^{1} and we have

lim inf_{k→+∞} ∫_{G} F (t, u_{k}(t), v_{k}(t)) dt ≥ ∫_{G} F (t, u_{0}(t), v_{0}(t)) dt.

Proof. Note that it is not restrictive to assume that lim inf_{k→+∞} I(u_{k}, v_{k}) = lim_{k→+∞} I(u_{k}, v_{k}) <

+∞ and sup_{k∈IN} I(u_{k}, v_{k}) = W < +∞.

Thus, the functions F_{k}(·) = F (·, u_{k}(·), v_{k}(·)), k ∈ IN are summable. Fixed an interval R_{0} = [a_{0}, b_{0}] ⊃ G, let us extend F_{k} to R_{0} by putting F_{k}(t) = 0, t ∈ R_{0} − G, k ∈ IN. Then let ϕ_{k} : R_{0} → IR, k ∈ IN, be the sequence defined by

ϕ_{k}(t) = ∫_{[a_{0},t]} F_{k}(τ) dτ.

Observe that (ϕ_{k})_{k∈IN} is a sequence of VAC functions which have equi-bounded variation in the sense of Vitali: in fact, for every finite partition D = [R] of the interval R_{0}, we have

∑_{R∈D} |Δ_{R} ϕ_{k}| = ∑_{R∈D} |∫_{R} ϕ_{k}(t) dt| ≤ ∫_{R_{0}} ϕ_{k}(t) dt + 2 ∫_{R_{0}} |λ(t)| dt ≤ W + 2 ∫_{G} |λ(t)| dt.

Thus, by virtue of Helly’s theorem ([20], see also [11c], proof of Theorem 1”), a function ϕ_{0} ∈ VBV exists such that (for a suitable subsequence)

ϕ_{k} ⟶ ϕ_{0} pointwise in R_{0}.

Taking into account of Proposition 3.8, we can deduce that the sequence of superficial derivatives

(D^{∗}ϕ_{k})_{k∈IN} satisfies (mv) on G.  (5.2)

It is easy to see that

D^{∗}ϕ_{k}(t) = F_{k}(t) a.e. in G, k ∈ IN  (5.3)

and moreover

(u_{k}, F_{k}, v_{k}) ∈ S1 ; _{Q_{0}}, k ∈ IN.  (5.4)

From (5.2) − (5.4) and the assumptions i), ii) and iv), by virtue of Theorem 4.1, we get that (u_{0}, D^{∗}ϕ_{0}, v_{0}) ∈ S1 ; _{Q_{0}} or equivalently

(u_{0}, v_{0}) ∈ S_{Q}^{1} and D^{∗}ϕ_{0}(t) ≥ F (t, u_{0}(t), v_{0}(t)) a.e. in G.  (5.5)

From assumption iii), we deduce that ϕ_{0}(t) ≥ λ̂(t), a.e. in R_{0}, where λ̂(t) = λ(t) on G, and λ̂(t) = 0 on R_{0} − G.

Let us consider the function Φ_{0}(t) = ϕ_{0}(t) − ∫_{[a_{0},t]} λ̂(τ) dτ, t ∈ R_{0}; since the interval function Δ_{R}Φ_{0} is additive and non-negative, we have ([22], Theorem III.1.28)

Δ_{R_{0}}Φ_{0} = ϕ_{0}(b) − ∫_{R_{0}} λ̂(t) dt ≥ ∫_{R_{0}} D^{∗}Φ_{0}(t) dt ≥ ∫_{G} D^{∗}ϕ_{0}(t) dt − ∫_{G} λ(t) dt.

Finally, from (5.5), we deduce that

I(u_{0}, v_{0}) ≤ ∫_{G} D^{∗}ϕ_{0}(t) dt ≤ ϕ_{0}(b) = lim_{k→+∞} ϕ_{k}(b) = lim_{k→+∞} I(u_{k}, v_{k})

which concludes the proof.

We wish to remark that, in the case both multifunction Q and integrand F (and hence multifunction Q̃_{0}) do not depend on variable x, then Theorem 5.1 reduces to the following result.

Theorem 5.2. (A lower semicontinuity result). Assume that A = G × A_{0} with A_{0} closed. Let (v_{k})_{k∈IN} be a sequence in S_{Q}^{1} and let v_{0} be a function in L_{1}(G, IR^{m}).

Suppose that

i) (v_{k})_{k≥0} satisfies (mv) on G.

ii) a function λ ∈ L_{1} exists such that F (t, v_{k}(t)) ≥ λ(t), a.e. in G, k ≥ 0;

iii) multifunction Q̃_{0} has property (Q) a.e. in G.

Then v_{0} ∈ S_{Q}^{1} and

lim inf_{k→+∞} ∫_{G} F (t, v_{k}(t)) dt ≥ ∫_{G} F (t, v_{0}(t)) dt.

Note that assumptions (Q) in Theorems 5.1 and 5.2 implies that multifunction Q is convex valued and the integrand F is convex in the last variable. For further remarks on the assumptions on the integrand, see Section 6.

Theorems 4.1 and 5.1 represent an extension of the closure and lower semicontinuity results in [11a,b,c,d] which are given in the particular case v_{k} = Du_{k}, k ≥ 0. To point out the interest of the present research we refer to Sections 7 and 8 where various new applications are presented.

Moreover, we wish to mention that the present paper is inspired by the results given in [9c,d,12] for control problems. Our theorems can be considered as a translation of those results in BV setting. Note that here a weaker topology is taken into consideration; more precisely, in [9c,d,12] weak-L_{1} convergence of the differential elements is adopted, whereas the present theorems do not require any convergence assumptions on the differential elements of the highest order (see Section 7 for details).

6. Some remarks on assumptions (Q) and (wF̃)

In order to illustrate the operativity of our semicontinuity result, we shall present some conditions on the integrand F which assure that the assumptions on multifunction Q̃_{0} are satisfied. Let us start by recalling some well-known results (see [9e] for details).

Property (Q) is intermediate between Kuratowski condition (K) and upper semicontinuity. As we already observed in [11b,c], assumption (Q) can not be replaced by weaker condition (K) in the present setting.

We recall that if Q̃_{0} satisfies property (Q) at the point (t_{0}, x_{0}), then Q̃_{0}(t_{0}, x_{0}) is a closed and convex set. Let us assume now that Q(t, x) = IR^{n}, (t, x) ∈ A, thus Q̃_{0}(t, x) = epi F (t, x, ·), (t, x) ∈ A. In this case Q̃_{0}(t_{0}, x_{0}) is closed and convex iff F (t_{0}, x_{0}, ·) is lower semicontinuous and convex. Moreover, the following result holds [9e].

Proposition 6.1. Multifunction Q̃_{0}(t, x) = epi F (t, x, ·), (t, x) ∈ A, satisfies property (Q) at the point (t_{0}, x_{0}) ∈ A iff F is seminormal at the same point, i.e.

(s/n) for every ε > 0 and y_{0} ∈ IR^{m}, a constant σ = σ(t_{0}, x_{0}, y_{0}, ε) > 0 and an affine function z : IR^{m} → IR exist such that F (t_{0}, x_{0}, y_{0}) < z(y_{0}) + ε F (t, x, y) ≥ z(y), for every y ∈ IR^{m} and (t, x) ∈ A with |t−t_{0}| < σ, |x−x_{0}| < σ.

For further conditions assuring property (Q) we refer to [9e], we only recall here the following criteria for seminormality (see [9b, 28, 27]). For more recent results about seminormality, see [2].

Proposition 6.2. Assume that F : A × IR^{m} → IR_{0}^{+} is continuous and F (t, x, ·) is convex, (t, x) ∈ A. Given a point (t_{0}, x_{0}) ∈ A, if there exists an affine function w : IR^{m} → IR such that

F (t_{0}, x_{0}, y) ≥ w(y), y ∈ IR^{m} and lim_{|y|→+∞}[F (t_{0}, x_{0}, y) − w(y)] = +∞

then F is seminormal at the point (t_{0}, x_{0}).

Proposition 6.3. Assume that F : A × IR^{m} → IR_{0}^{+} is continuous and F (t, x, ·) is convex, (t, x) ∈ A. Then the function F_{ε} : A × IR^{m} → IR_{0}^{+}, ε > 0 defined by

F_{ε}(t, x, y) = F (t, x, y) + ε|y|

is seminormal at every point (t_{0}, x_{0}) ∈ A.

Proof. Note that F_{ε} satisfies the assumptions of Proposition 6.2 with z(y) = 0, y ∈ IR^{m}.

Condition (wF̃) acts on the variable x; indeed, as we already observed, it is trivial provided Q̃_{0} does not depend on x. It was introduced in [11d] as a weakening of conditions (F′_{i}), i = 1, …, 3 in [11c]. They can be considered as variants of Cesari’s Lipschitz-type property (F) ([9e]) which was proposed in Cesari-Suryanarayana [13] as an important tool in optimal control theory. The Authors also showed that these assumptions are rather natural, easy to verify and actually satisfied in many applications. See also Cesari-Angell [10].

In what follows we assume that A = G × A_{0}, with A_{0} closed, and Q : G → 2^{IR^{n}}. Moreover, let S = (u_{k}, v_{k})_{k∈IN} be a sequence in S_{Q}^{1} and let u_{0} ∈ L_{1}(G, IR^{n}) be such that u_{k} L_{1}–converges to u_{0}.

The main consequence of condition (F) is the following result (see [13a])

Proposition 6.4. Suppose that F satisfies conditions (C) for every ε > 0 a compact set K_{ε} ⊂ G exists such that meas (G − K_{ε}) < ε, and

F_{/K_{ε}×A_{0}×IR^{m}} is continuous;

(F) for a.e. t ∈ G, every u_{1}, u_{2} ∈ A_{0} and every k ∈ IN,

|F (t, u_{1}, v_{k}(t)) − F (t, u_{2}, v_{k}(t))| ≤ C ϕ(|u_{1} − u_{2}|)

where C > 0 is a constant and ϕ : IR_{0}^{+} → IR_{0}^{+} is a non decreasing function such that ϕ(0+) = 0, ϕ(ζ) ≤ c|ζ|, c ≥ 0, for all ζ ≥ ζ_{0} ≥ 0.

Then

lim_{k→+∞} ∫_{G} |F (t, u_{k}(t), v_{k}(t)) − F (t, u_{0}(t), v_{k}(t))| dt = 0.  (D)

We called our condition (wF̃) = weak (F) since we thought that it could be implied by assumption (F) even if we were not able to prove this result. Now we can finally justify this terminology (see Propositions 6.5 and 6.7 below).

Proposition 6.5. If F satisfies conditions (C) and (F), then (R) for every ε > 0, every compact set K ⊂ G with meas (G − K) < ε and a.e.

t_{0} ∈ K

lim sup_{h→0} lim sup_{k→+∞} |q_{h}|^{−1} ∫_{q_{h}−K}[F (t, u_{0}(t_{0}), v_{k}(t)) − F (t, u_{k}(t), v_{k}(t))] dt ≤ 0.

P. Brandi, A. Salvadori / On lower semicontinuity in BV setting 163

Proof. By virtue of Proposition 6.4

δ_{k}(·) = F (·, u_{0}(·), v_{k}(·)) − F (·, u_{k}(·), v_{k}(·)) ⟶ 0 in L_{1}

then we deduce that (see also Proposition 3.3), for a.e. t_{0} ∈ G

lim_{h→0} lim_{k→+∞} − ∫_{q_{h}} |δ_{k}(t)| dt = 0.  (6.1)

Let ε > 0 be fixed and let K ⊂ G be a compact set with meas (G − K) < ε. By virtue of well-known density results [19], for a.e. t_{0} ∈ K

lim_{h→0} − ∫_{q_{h}} |u_{0}(t) − u_{0}(t_{0})| dt = 0  lim_{h→0} \frac{meas (q_{h} − K)}{|q_{h}|} = 0.  (6.2)

Let t_{0} ∈ K be fixed in such a way that assumption (F) and (6.2) hold. Denoted by E = {t ∈ G : |u_{0}(t_{0}) − u_{0}(t)| ≥ ζ_{0}}, from (F) we have

|q_{h}|^{−1} ∫_{q_{h}−K}[F (t, u_{0}(t_{0}), v_{k}(t)) − F (t, u_{0}(t), v_{k}(t))] dt ≤
≤\frac{C}{|q_{h}|} ∫_{q_{h}−K} ϕ(|u_{0}(t_{0}) − u_{0}(t)|) dt ≤
≤\frac{Cc}{|q_{h}|} ∫_{(q_{h}−K)∩E} |u_{0}(t_{0}) − u_{0}(t)| dt + \frac{C}{|q_{h}|} ∫_{(q_{h}−K)−E} ϕ(ζ_{0}) ≤
≤Cc − ∫_{q_{h}} |u_{0}(t_{0}) − u_{0}(t)| dt + Cϕ(ζ_{0})\frac{meas (q_{h} − K)}{|q_{h}|}

and hence, from (6.2) we deduce

lim sup_{h→0} lim sup_{k→+∞} |q_{h}|^{−1} ∫_{q_{h}−K}[F (t, u_{0}(t_{0}), v_{k}(t)) − F (t, u_{k}(t), v_{k}(t))] dt ≤ 0.  (6.3)

The assertion is an immediate consequence of (6.1) and (6.3).

Remark 6.6. Let us consider the following weakening of condition (R) (wR) for every ε, ϱ > 0, every compact set K ⊂ G with meas (G − K) < ϱ and

a.e. t_{0} ∈ K, there is a constant 0 < h′ = h′(ε, ϱ, K, t_{0}, u_{0}(t_{0}), S) such that for a.e. 0 < h < h′ a subsequence (s_{k})_{k∈IN} exists with the property that

|q_{h}|^{−1} ∫_{q_{h}−K}[F (t, u_{0}(t_{0}), v_{s}_{k}(t)) − F (t, u_{s}_{k}(t), v_{s}_{k}(t))] dt ≤ ε.

Let us present now a criterion for (wF̃).

Proposition 6.7. If F satisfies condition (wR), then Q̃_{0} satisfies condition (wF̃), with respect to S, at (t_{0}, u_{0}(t_{0})) for a.e. t_{0} ∈ G.

Proof. From the assumption u_{k} L_{1}–converges to u_{0}, we deduce that there is a subsequence, say still (k), such that u_{k} → u_{0} a.e. in G. Then, by virtue of Egoroff’s and Lusin’s theorems, for every ε > 0, a compact set K_{ε} exists such that meas (G − K_{ε}) < ε and the following conditions hold

u_{0}_{/K_{ε}} is continuous  (6.4)
u_{k} → u_{0} uniformly on K_{ε}.  (6.5)

Let ε > 0 be fixed. We consider the sequence ε_{m} = \frac{ε}{2m}, m ∈ IN. Let K_{m} denote the compact set corresponding to ε_{m} and let N_{m} ⊂ K_{m} be the null subset of the points t_{0} to which (wR) does not apply; note that N = ∪_{m} N_{m} ∪(∩_{m}(G − K_{m})) is a null set.

Let t_{0} ∈ G − N be fixed. Then there exists \overline{m} = \overline{m}(t_{0}) ∈ IN such that t_{0} ∈ K\overline{m} − N\overline{m}. From (6.4) and (6.5) we deduce that a constant \overline{h} = \overline{h}(ε, t_{0}, u_{0}) > 0 and an integer \overline{k} = \overline{k}(ε, (u_{k})_{k≥0}) exist such that for every 0 < h < h_{0} and every k > \overline{k}, if t ∈ K\overline{m} ∩ q_{h}

|u_{k}(t) − u_{0}(t_{0})| < |u_{k}(t) − u_{0}(t)| + |u_{0}(t) − u_{0}(t_{0})| < ε\overline{m} + ε\overline{m} < \frac{ε}{2} + \frac{ε}{2} = ε. (6.6)

Moreover, let 0 < h′ = h′(ε, t_{0}, u_{0}(t_{0}), S) ≤ ε be a constant such that (see Remark 6.6) for a.e. 0 < h < min(\overline{h}, h′) a subsequence (s_{k})_{k∈IN} exists with the property that s_{k} ≥ \overline{k}, k ∈ IN, and

|q_{h}|^{−1} ∫_{q_{h}−K\overline{m}}[F (t, u_{0}(t_{0}), v_{s}_{k}(t)) − F (t, u_{s}_{k}(t), v_{s}_{k}(t))] dt ≤ ε\overline{m} < ε  (6.7).

Now, let us consider the sequence (\overline{u}_{s_{k}}, w_{s_{k}}^{0}, w_{s_{k}})_{k∈IN} defined by

\overline{u}_{s_{k}} = \{ \begin{matrix}u_{0}(t_{0}) & t ∈ q_{h} − K\overline{m} \\ u_{s_{k}}(t) & t ∈ q_{h} ∩ K\overline{m} \end{matrix}

w_{s_{k}}(t) = v_{s_{k}}(t)  w_{s_{k}}^{0}(t) = F (t, \overline{u}_{s_{k}}(t), v_{s_{k}}(t))  t ∈ G.

Of course the sequence (\overline{u}_{s_{k}}, w_{s_{k}}^{0}, w_{s_{k}})_{k∈IN} lies in S_{Q_{0}} and moreover, from (6.6) and (6.7), we get

|\overline{u}_{s_{k}}(t) − u_{0}(t_{0})| ≤ ε t ∈ q_{h}

\begin{aligned}− ∫_{q_{h}}[w_{s_{k}}^{0}(t) − f(t, u_{s_{k}} & (t), v_{s_{k}}(t))] dt = \\ & = |q_{h}|^{−1} ∫_{q_{h}−K\overline{m}}[F (t, u_{0}(t_{0}), v_{s}_{k}(t)) − F (t, u_{s}_{k}(t), v_{s}_{k}(t))] dt ≤ ε \end{aligned}

which proves the assertion.

P. Brandi, A. Salvadori / On lower semicontinuity in BV setting 165

Let us prove another sufficient condition for (R).

Proposition 6.8. If F satisfies the condition (F′′) two functions Φ : A_{0} → IR and ψ ∈ L_{1} exist such that Φ(u_{k}) converges to

Φ(u_{0}) in L_{1} and for a.e. t_{0} ∈ G a constant ρ = ρ(t_{0}, u_{0}(t_{0}), S) > 0 exists such

that for a.e. t ∈ G with |t − t_{0}| < ρ and k ∈ IN we have

|F (t, u_{0}(t_{0}), v_{k}(t)) − F (t, u_{k}(t), v_{k}(t))| ≤ |Φ(u_{0}(t_{0})) − Φ(u_{k}(t))| + |ψ(t_{0}) − ψ(t)|,

then F satisfies condition (R).

Proof. Observe that, by virtue of well-known density results [19], for a.e. t_{0} ∈ G we have

lim_{h→0} − ∫_{q_{h}} |ψ(t_{0}) − ψ(t)| dt = 0,  lim_{h→0} − ∫_{q_{h}} |Φ(u_{0}(t_{0})) − Φ(u_{0}(t))| dt = 0  (6.8)
and  lim_{h→0} lim_{k→∞} − ∫_{q_{h}} |Φ(u_{0}(t)) − Φ(u_{k}(t))| dt = 0.

Let t_{0} ∈ G be fixed in such a way that (6.8) and (F′′) are satisfied, then

\begin{aligned}& − ∫_{q_{h}} [F (t, u_{0}(t_{0}), v_{k}(t)) − F (t, u_{k}(t), v_{k}(t))] dt ≤ \\ ≤ & − ∫_{q_{h}} |Φ(u_{0}(t_{0})) − Φ(u_{0}(t))| dt + − ∫_{q_{h}} |Φ(u_{0}(t)) − Φ(u_{k}(t))| dt + − ∫_{q_{h}} |ψ(t_{0}) − ψ(t)| dt \end{aligned}

and the assertion follows by virtue of (6.8).

Remark 6.9. Note that, by virtue of Propositions 6.5 and 6.8, Proposition 6.7 applies to integrands of the type

1) F (t, ·, y) Lipschitzian, t ∈ G, y ∈ IR^{m};

2) F (t, x, y) = f(t, x, y)+F_{0}(t, y) with f, F_{0} Carathéodory and f satisfying assumption (F);

3) F (t, x, y) = f(t, x, y)+F_{0}(t, y) with F_{0} Carathéodory and f satisfying assumption (F′′);

4) F (t, x, y) = |f(t, x, y) + F_{0}(t, y)| with F_{0} and f as above. Moreover it is trivially satisfied in the case

5) F is bounded. In other words, the multifunction Q̃_{0} associated to any integrands of these classes satisfies condition (wF̃).

Finally, from Theorem 5.1 and Propositions 6.1, 6.3 and 6.7, the following lower semicontinuity result can be deduced.

Theorem 6.10. (A lower semicontinuity result) Assume that A_{0} ⊂ IR^{n} is closed. Let S = (u_{k}, v_{k})_{k≥0} be a sequence in L_{1}(IR^{n+m}) such that

i) u_{k} ∈ A_{0}, a.e. in G, k ∈ IN; ii) u_{k} L_{1}–converges to u_{0}; iii) (v_{k})_{k≥0} is L_{1}–equibounded and satisfies (mv).

Suppose that F : G × A_{0} × IR^{m} → IR_{0}^{+} is continuous, satisfies condition (wR) and F (x, y, ·) is convex, x ∈ A_{0} a.e. t ∈ G.

Then u_{0}(t) ∈ A_{0} a.e. in G and

lim inf_{k→+∞} ∫_{G} F (t, u_{k}(t), v_{k}(t)) dt ≥ ∫_{G} F (t, u_{0}(t), v_{0}(t)) dt.

Proof. For every ε > 0, let F_{ε} : G × A_{0} × IR^{m} → IR_{0}^{+} be the integrand defined by

F_{ε}(t, x, y) = F (t, x, y) + ε|y|.

By virtue of Propositions 6.3, 6.1 and 6.7, F_{ε} satisfies all the hypotheses of Theorem 5.1, thus

lim inf_{k→+∞} ∫_{G} F_{ε}(t, u_{k}(t), v_{k}(t)) dt ≥ ∫_{G} F_{ε}(t, u_{0}(t), v_{0}(t)) dt ≥ ∫_{G} F (t, u_{0}(t), v_{0}(t)) dt. (6.9)

Now

∫_{G} F_{ε}(t, u_{k}(t), v_{k}(t)) dt = ∫_{G} F (t, u_{k}(t), v_{k}(t)) dt + ε ∫_{G} |v_{k}(t)| dt, k ∈ IN

and, since (v_{k})_{k∈IN} is equibounded in L_{1}(G, IR^{m}), the assertion is an immediate consequence of (6.9).

Remark 6.11. We wish to point out that if F : G × A_{0} × IR^{m} → IR_{0}^{+} satisfies the assumptions of Theorem 8.1, then also the integrand F_{a} : G × A_{0} × IR^{m} → IR_{0}^{+} defined by

F_{a}(t, x, y) = F (t, x, ⟨a(t), y⟩)

with a : G → IR^{m} continuous, satisfies the same assumptions.

7. Some remarkable particular cases

We wish to point out that the following formulations can be framed as particular cases of the present setting. Let us consider a multiple integral of the type

I[x] = ∫_{G} F (t, (Ux)(t), (Lx)(t)) dt  (7.1)

subjected to the constraints

(t, (Ux)(t)) ∈ A, (Lx)(t) ∈ Q(t, (Ux)(t)) a.e. in G  (7.2)

where U and L are given operators and Q is a given multifunction. Note that (7.1) − (7.2) represents a rather abstract version of integral functional of the calculus of variations with constraints on the differential elements. We recall that [9e] every optimal control problem can be reduced to a suitable problem of the calculus of variations with constraints on the gradient. By virtue of this remark, (1.1) − (1.2) contains also the following formulation: a cost functional

I[x, y] = ∫_{G} f_{0}(t, (Ux)(t), y(t)) dt  (7.3)

with state equation

(Lx)(t) = f(t, (Ux)(t), y(t)) a.e. in G  (7.4)

and possible constraints of the form

(t, (Ux)(t)) ∈ A , (Lx)(t) ∈ Q(t, (Ux)(t)) a.e. in G.  (7.5)

Note that operators U appearing in the definition of I and in the constraints need not be the same, but may be thought of as different components of the same operator. Analogous for operator L.

Optimal control problems of type (7.3) − (7.5) were studied by Cesari [9c,d] and Cesari-Cowles [12], who also gave closure and lower closure results with respect to the weak topology in Sobolev’s spaces. Forthermore, lower semicontinuity theorems for integral functionals of type (7.1) had already been proved by Rothe [23] and Fichera [18], again in the same setting.

The aim of the present paper is to study the lower semicontinuity in BV setting and, as we shall show in what follows, we are able to prove the lower semicontinuity of (7.1) − (7.2) (and hence (7.3) − (7.5)) with respect to a weaker topology which do not require any convergence assumptions on the differential elements of the highest order. Before to enter in details, we wish to mention that the interest for formulation (7.1)−(7.2) has recently increased since optimization problems of this type, with BV optimal solutions, were adopted as a mathematical model to describe the plastic deformation of beams and plates under different loads [11e,14,3a,b,4].

Let (S, w) be a topological space and let U : S → L_{1}(G, IR^{n}) and L : S → L_{1}(G, IR^{m}) be given operators. Our lower semicontinuity results apply under the assumption that

if u_{k} w-converges to u_{0}, then (for a suitable subsequence) we have that Uu_{k} L_{1}–converges to Uu_{0} and (Lu_{k})_{k≥0} is L_{1}–equibounded and satisfies (mv).

For example, the following particular cases can be framed in our theory (see Propositions 3.2, 3.3, 3.7, 3.10 and Theorem 4.1):

(a) (x_{k})_{k∈IN} is a sequence in W^{1,1}(G, IR^{N})L_{1}–converging to a function u_{0} ∈ BV (G, IR^{N})

and

Ux = x, Lx = Dx or Lx = div x;

(b) (x_{k})_{k≥0} is the same sequence as in (a) and

(Ux)(t) = Φ(t, x(t)), (Lx)(t) = D[Ψ(t, x(t))]

where Φ and Ψ satisfies the assumptions of Corollary 3.6 and Proposition 3.10 respectively;

(c) (x_{k})_{k∈IN} is a sequence in W^{p+1,1}(G, IR^{N}) which W^{p,1}–converges to a function u_{0} ∈ W^{p,1}(G, IR^{N}), with D^{p}u_{0} ∈ BV (G, IR^{M}), and

Ux = (x, Dx, D^{2}x, ..., D^{p}x), Lx = D^{p+1}x

where D^{2}x denotes the essential gradients of Dx and so on;

(d) (x_{k})_{k≥0} is the same sequence as in (c) and

U is any operator on the variables (x, Dx, D^{2}x, ..., D^{p}x) which is continuous with respect to L_{1}−topology, L is a linear combination of the elements of D^{p+1}x, for example Lx = div (D^{p}x);

(e) (x_{k})_{k≥0} is the same sequence as in (c) and

(Ux)(t) = Φ(t, X(t)), (Lx)(t) = D[Ψ(t, X(t))]

where X = (x, Dx, D^{2}x, ..., D^{p}x) and Φ and Ψ are functions as in (b).

Finally we wish to mention that also the following operators can be framed in our setting

(f) (x_{k})_{k≥0} is a sequence in L_{1}(G, IR^{n}) and

(Ux)(t) = ∫_{G} K(t, τ) x(τ) dτ, Lx = x

where K is a suitable kernel (see [18] and [23,24]).

Remark 7.1. Existence results for problems of the calculus of variations can be derived from the closure and lower semicontinuity statements of Sections 4 an 5. More precisely, let us assume that functional I is defined on W^{1,1}(G, IR^{N}) and let us consider the Serrin-type extension to BV (G, IR^{N})

J[x] = inf_{Γ(x)} lim inf_{k→+∞} I[x_{k}]

where Γ(x) = {(x_{k})_{k∈IN} : x_{k} ∈ W^{1,1}(G, IR^{N}), satisfies (7.2) and x_{k} L_{1}–converges to x} ,

if Γ(x) = ϕ we put J[x] = +∞, as usual.

An analogous functional can be defined in the case I ranges over W^{p,1}(G, IR^{N}).

By virtue of our lower semicontinuity results, we can deduce that the natural relation between I and J occurs

J[x] = I[x] on W^{1,1}(G, IR^{N}) and J[x] ≥ I[x] on BV (G, IR^{N}).

Moreover, by means of a standard process, we can prove the existence of minima for functional J over a class Ω satisfying suitable compactness conditions. We do not enter into detail here, but only refer to [9e] for the classical theory and to [11b,c,e] for more recent developments.

8. Applications to a class of integral functionals

Let us consider now functionals of the type

I[x] = ∫_{G}|⟨a(t), D[Ψ(t, x(t))]⟩ + Φ(t, x(t))| dt

where a : G → IR^{νN}, Φ : A → IR and Ψ : A → IR^{N} are given functions. This class of functionals takes its source from problems of conservation laws. Cesari [9f,g,h] discussed the related optimization problems under the assumptions that Φ is continuous and Ψ is of class C^{1}. In order to get the existence of BV minima for the Serrin-type extension of I, Cesari adopted a suitable transformation which allows to reduce functional I to a standard integral of the type

I^{∗}[x^{∗}] = ∫_{G} F (t, x^{∗}(t), Dx^{∗}(t)) dt

to which the existence results of [11c,d] apply.

Now, by virtue of the present formulation where abstract operators are involved, we can deal with functional I directly, with remarkable advantages both in the assumptions and in the proof. We only mention here the lower semicontinuity result since, as we already mentioned (see Remark 6.6), the existence theorem is then a standard consequence.

Theorem 8.1. Assume that A = G × A_{0} with A_{0} closed. Let (x_{k})_{k∈IN} be a sequence in W^{1,1}(G, IR^{n}) and let x_{0} ∈ BV (G, IR^{n}) be such that

i) x_{k}(t) ∈ A_{0} a.e. in G, k ∈ IN; ii) (x_{k})_{k∈IN} has equibounded variation and L_{1}–converges to x_{0}.

Suppose that iii) a is continuous, iv) Ψ is Lipschitzian,

v) Φ is Carathéodory and there exists ϕ ∈ L_{1} such that |Φ(t, x_{k}(t))| ≤ ϕ(t) a.e. in G, k ∈ IN .

Then the limit function satisfies x_{0}(t) ∈ A_{0} a.e. in G, and we have

lim inf_{k→+∞} I[x_{k}] ≥ I[x_{0}].

Proof. Let U and L be the operators defined by

(Ux)(t) = Φ(t, x(t)), (Lx)(t) = D[Ψ(t, x(t))].

By virtue of Proposition 3.10 and Corollary 3.6, we have that (for a suitable subsequence)

Ux_{k} → Ux_{0} in L_{1} and (Lx_{k})_{k∈IN} satisfies (mv) on G.

Moreover, it is easy to see that the sequence (D[Ψ(·, x_{k}(·))])_{k∈IN} has equibounded variation, thus (Lx_{k})_{k∈IN} is bounded in L_{1}(G, IR^{νN}).

Let us consider the integrand F : G × A_{0} × IR^{νN} → IR_{0}^{+} defined by

F (t, x, y) = |⟨a(t), y⟩ + x|

of course it satisfies assumption of Theorem 8.1 (see also Remark 7.1) and the assertion follows as an immediate application of Proposition 6.5.

Remark 8.2. Note that a result analogous to that of Theorem 8.1 can be proved for the functional

I[x] = ∫_{G}[⟨a(t), D[Ψ(t, x(t))]⟩ + Φ(t, x(t))]^{+} dt.

We recall that the lower semicontinuity for integrals of the type

I[x] = ∫_{G}[⟨a(t, x(t)), Dx(t)⟩ + b(t, x(t))]^{+} dt

with respect to weak topology in Sobolew’s spaces, was discussed in [17,1].

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