Journal of Convex Analysis Volume 4 (1997), No. 1, 129–147

Non-Uniform Integrability and Generalized Young Measures

J.J. Alibert, G. Bouchitté

Laboratoire ANLA, UFR Science, Université de Toulon et du Var, BP 132, 83957 La Garde Cedex, France.

Received September 6, 1995 Revised manuscript received February 27, 1996

Given a bounded sequence (u_{n}) in L1(Ω, μ; IR^{d}), we describe the weak limits in the sense of measures of f(x, u_{n}) μ for a class of continuous integrands with linear growth at infinity. The defect of uniform integrability of the sequence f(x, u_{n}) is described by a measure m and a family of probability measures on S^{d−1} whereas the classical Young measure is associated with the biting limits in the sense of Chacon’s lemma. Some consequences of this new approach are given in Calculus of Variations.

1. Introduction

The oscillatory properties of a weakly convergent sequence (u_{n}) of functions in L^{1}(Ω, μ, IR^{d}) can be very well described by the parametrized measure (or Young measure) it generates. This parametrized measure is a family of probabilities (ν_{x})_{x∈Ω} on IR^{d} such that:

f(x, u_{n}(x)) ⇀ ∫_{IR_{d}} f(x, z)ν_{x}(dz)

for every Caratheodory function f : Ω × IR^{d} → IR such that f(x, u_{n}(x)) is weakly convergent in L^{1} (see [1,2,3]). In practice this representation formula is applied in case (u_{n}) is bounded in some L^{p} and f satisfies a suitable growth condition: |f(x, z)| ≤ C(1 + |z|^{q}) (1≤ q < p). In many situations f(x, u_{n}(x)) represents a density of energy and is bounded in L^{1}(Ω) but non uniformly integrable (problems in Plasticity and fracture mechanics). A way to overcome this difficulty already used in Calculus of Variations was to replace the weak convergence in L^{1} by the convergence in the biting sense introduced by Gaposhkin and Chacon (see for example [4]). In this case, one can construct also a parametrized measure denoted “biting Young measure” by Kinderlehrer and Pedregal [5] (see also [2]). In fact in this approach, we have no information on the singular part of the weak limit of energies f(x, u_{n}(x)) μ in the sense of Radon measures. Note also that the regular part of this weak limit do not coincide in general with the biting limit (see Ball and Murat [3] or Example 3.2 in this paper). An important precursor to the present paper is the work by Di Perna and Majda [6] where an extension of the concept of Young measure was introduced in

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

130 J.J.Alibert, G. Bouchitté / Non-uniform integrability and generalized young measures

order to study concentrations effects for measure-valued solutions of incompressible fluid equations (see also the notion of defect measure introduced in [7]) .

Here we consider bounded sequences (u_{n}) in L^{1}(Ω, μ, IR^{d}) which we see as bounded μabsolutely continuous vector measures. Using a slicing argument for Radon measures constructed on the product Ω × B̄_{d} where B̄_{d} denotes the closed unit ball of IR^{d} (seen as the compactification of IR^{d}), we obtain an integral representation formula for the limit λ of f(x, u_{n}(x)) μ of the form:

λ = (∫_{IR_{d}} f(x, z)ν_{x}(dz)) μ + (∫_{S^{d−1}} f^{∞}(x, z)ν_{x}^{∞}(dz)) m ,

where:

− ν_{x} is a probability measure on IR^{d},

− m is a Radon measure on Ω,

− ν_{x}^{∞} is a probability measure on S^{d−1}the unit sphere of IR^{d},

− f(x, z) belongs to a suitable class F(Ω × IR^{d}) of continuous functions such that

\frac{f(x, z)}{1 + |z|} is bounded and uniformly continuous with respect to z and

f^{∞}(x, z) := lim_{z′→z,|z′|<1}(1 − |z′|)f(x, \frac{z′}{1 − |z′|}) = lim_{t→∞} \frac{f(x, tz)}{t} .

The paper is organized as follows: in section 2, we fix the notations and state the main theorem (Theorem 2.5) and its corollary (Theorem 2.9) where we evaluate the biting limit of the sequence f(x, u_{n}(x)) in term of the probability measure ν_{x} and characterize the weak compactness property in L_{loc}^{1} in term of the measures m and ν_{x}^{∞}. Proofs are given in section 4. In section 3, we identify the probability measures ν_{x} and ν_{x}^{∞} on some relevant examples exhibiting concentrations and oscillations effects. In section 5, we present some applications in Calculus of variations: we give in Theorem 5.1 an improved version of a lower semicontinuity result for convex functional on measures (see [8],[9],[10]) and in Theorem 5.3, we recover and precise a recent result of strong convergence in L^{1} under strict convexity obtained by Brezis [11].

2. Notations and main results

In all the sequel Ω denotes either an open subset or a compact subset of IR^{N}, |x| the Euclidian norm of x ϵ IR^{N} , B(Ω) the σ -algebra of Borel subsets of Ω , 1_{B} the characteristic function of some Borel set B. We will denote by B_{d} the unit open ball of IR^{d} , by B̄_{d} its closure. S^{d−1} denotes the unit sphere of IR^{d}. Let us introduce also the following notations:

- M(Ω, IR^{d}) is the space of all vector valued Borel measures λ : B(Ω) → IR^{d} with finite variation on Ω. Recall the variation of λ on B is defined by:

|λ|(B) := sup{∑_{k} |λ(B_{k})| : ⋃_{k} B_{k} ⊂ B, B_{k} pairwise disjoints}

J.J.Alibert, G. Bouchitté / Non-uniform integrability and generalized young measures 131

- The space M(Ω; IR^{d}) is naturally in duality with the space C_{0}(Ω; IR^{d}) of all continuous functions of Ω to IR^{d} vanishing at infinity and we will consider the associated weak* topology. The weak* convergence of a sequence (λ_{n}) in M(Ω; IR^{d}) will be denoted λ_{n} ⇀^{∗} λ which means:

∫_{Ω} φ(x)·\frac{dλ_{n}}{d|λ_{n}|}(x) |λ_{n}|(dx) → ∫_{Ω} φ(x)·\frac{dλ}{d|λ|}(x) |λ|(dx) for every φ ∈ C_{0}(Ω; IR^{d}) (2.1)

When (2.1) holds for every φ in the space C_{b}(Ω; IR^{d}) of bounded continuous functions of Ω to IR^{d}, we will say that the convergence of (λ_{n}) is tight. Recall that it is the case when the following condition is satisfied (automatic if Ω is compact):

∀ε > 0 ,  ∃K compact ⊂ Ω :  sup_{n} |λ_{n}(Ω\K)| < ε  (tightness)

- Given a Radon measure m on Ω and a topological space X, we denote by P(Ω, m, X) the set of families (ν_{x})_{x∈Ω} of probability measures on X (Young family) which depend m-measurably on x. Sometimes the integral ∫_{X} f(z)ν_{x}(dz) will be denoted simply ⟨ν_{x}, f⟩ while in case X is imbedded in a linear space , [ν_{x}] will denote the barycenter of ν_{x} (corresponding to f(z) = z when ∫ |z|ν_{x}(dz) is finite).

- For every function g on Ω × IR^{d} , we define the recession function g^{∞} by:

g^{∞}(x, z) := lim inf_{t→∞,z′→z} \frac{g(x, tz′)}{t} ,  (2.2)

which is clearly lower semicontinuous and positively 1-homogeneous with respect to z. The definition of the recession cone of a subset F of IR^{d} is deduced by taking the indicator function of F:

F^{∞} := {z ϵ IR^{d}; ∃z_{n} → z ∃ t_{n} → ∞ t_{n}z_{n} ϵ F } .  (2.3)

- F(Ω × IR^{d}) is defined as the class of continuous functions f on Ω × IR^{d} such that the mapping: (x, z) ∈Ω × B_{d} → (1 − |z|)f(x, \frac{z}{1 − |z|}) can be extended into a bounded

continuous function on Ω × B̄_{d} . Denoting by f̃ this (unique) extension, we have:

∃Λ > 0 :  |f(x, z)| ≤ Λ(1 + |z|)  for all (x, z) ∈ Ω × IR^{d} ,  (2.4)
f̃(x, z) := \{ \begin{matrix}(1 − |z|)f(x, \frac{z}{1 − |z|}) & if |z| < 1 , \\ f^{∞}(x, z) & if |z| = 1 . \end{matrix}  (2.5)

Note that for such a function f , the liminf in the definition (2.2) is actually a limit (this by the continuity of f̃(x, .) on B̄_{d}). Moreover F(Ω × IR^{d}) endowed with the

norm sup\{\frac{|f(x, z)|}{1 + |z|}; (x, z) ∈ Ω × IR^{d}\} is a Banach space.

Remark 2.1. In [6], Di Perna and Majda consider also a particular class of admissible integrands f which satisfy a growth condition of order p. They give a representation formula for the weak ∗ limit of f(x, u_{n}(x)) μ when (u_{n}) is bounded in L^{p}(Ω, μ; IR^{d}). This formula has a concrete form when f(x, .) is positively homogeneous of order p or when \frac{f(x,z)}{1+|z|^{p}} vanishes as |z| → ∞. Let us notice that the latter cases are recovered from our

results (related to the case p = 1) by setting: v_{n} := |u_{n}|^{p−1}u_{n} and g(x, z) := f(x, |z|^{\frac{1−p}{p}} z). Indeed we can write f(x, u_{n}(x)) = g(x, v_{n}(x)) where the new sequence (v_{n}) is bounded in L^{1}(Ω, μ; IR^{d}) and g belongs to F(Ω × IR^{d}).

Some properties related to the class F(Ω × IR^{d}) are collected in the two lemmas below (proved in the appendix).

Lemma 2.2. Let f be continuous on Ω × IR^{d} and satisfy (2.4). Then f belongs to F(Ω × IR^{d}) in the following cases:

(i) f is bounded ( in that case f^{∞} ≡ 0), (ii) f(x, λz) = λf(x, z) for every λ ≥ 0 and (x, z) ϵ Ω × IR^{d} (in that case f^{∞} = f), (iii) f is convex (or concave) with respect to z and f^{∞} is continuous (for example f inde-

pendent of x).

Lemma 2.3. Let f be lower semicontinuous on Ω × IR^{d} such that for some suitable Λ > 0: f(x, z) ≥ −Λ (1 + |z|) for every (x, z) ∈ Ω × IR^{d}. Define:

h_{f}(x, z) := lim inf\{ \frac{f(x′, tz′)}{t} ; t → ∞, x′ → x, z′ → z\}  (2.6)

Then there exists a non decreasing sequence (f_{p}) in F(Ω × IR^{d}) such that:

f_{p}(x, z) ≥ −Λ (1 + |z|) , supf_{p} = f Moreover for any such a sequence (f_{p}), we have: supf_{p}^{∞}(x, z) = h_{f}(x, z).

Remark 2.4. The integrand h_{f} defined by (2.6) looks as a l.s.c. regularization of f^{∞} with respect to x and h_{f}(x, .) = f^{∞}(x, .) for every x when f has a good behaviour (for example f ∈ F(Ω × IR^{d})). In particular it is the case when f satisfies the assumptions of the lemma, is convex with respect to z and f(.,0) is locally bounded (see the proof in Appendix).

In all the sequel, μ is a given non negative Radon measure on Ω. We are in the position to state our main result:

Theorem 2.5. Let (u_{n}) be a bounded sequence in L^{1}(Ω, μ; IR^{d}) and F a closed subset of IR^{d} such that u_{n}(x) ∈ F μ a.e.. Then there exists a subsequence (u_{n_{k}}), a nonnegative Radon measure m and parametrized families of probabilities (ν_{x}) ∈ P(Ω, μ; IR^{d}) , (ν_{x}^{∞}) ∈ P(Ω, m; S^{d−1}) such that:

f(x, u_{n_{k}}(x)) μ ⇀^{∗} ⟨ν_{x}, f(x, .)⟩ μ + ⟨ν_{x}^{∞}, f^{∞}(x, .)⟩ m

for every f in F(Ω × IR^{d}).

Moreover the following properties hold:

(i) ∫_{Ω}⟨ν_{x}, |z|⟩dμ < +∞ and supp ν_{x} ⊂ F for μ a.e. x ϵ Ω . (ii) supp m ⊂ supp μ and supp ν_{x}^{∞} ⊂ F^{∞} ∩ S^{d−1} for m a.e. x ∈ Ω . (iii) For every f satisfying the conditions of Lemma 2.3:

lim inf_{k→∞} ∫_{Ω} f(x, u_{n_{k}}) dμ ≥ ∫_{Ω}⟨ν_{x}, f(x, .)⟩ dμ + ∫_{Ω}⟨ν_{x}^{∞}, h_{f}(x, .)⟩ dm

where h_{f} is defined as in (2.6).

Remark 2.6. a) In fact, it is easy to check that the Theorem 2.5 can be reformulated assuming only that the initial sequence (u_{n}) is bounded in L_{loc}^{1}(Ω, μ; IR^{d}) . In that case the weak ∗ convergence has to be understood in the sense that (2.1) holds only for those functions φ in C_{0}(Ω; IR^{d}) which are compactly supported in Ω.

b) Using the argument in Remark 2.10 below, it is possible to prove that the assertion (i) of the theorem still holds if we replace F by any μ- measurable closed valued multifunction F(x).

Now we want to discuss the connections of Theorem 2.5 with the weak convergence in L^{1} and also with the Chacon biting’s lemma [3]. Let us recall briefly the main features of the latter convergence which has been used in many situations where weak compactness in L^{1} fails. Recall that, by the Dunford-Pettis theorem, a bounded sequence (u_{n}) in L_{μ}^{1}(Ω) (μ(Ω) being finite) is weakly relatively compact if and only if it is uniformly μ-integrable, that is:

lim_{R→∞} lim sup_{n→∞} ∫_{|u_{n}|≥R} |u_{n}|dμ = 0.

Lemma 2.7. (Chacon) Assume that μ(Ω) is finite and let (u_{n}) be a bounded sequence in L^{1}(Ω, μ; IR^{d}). Then there exist a function u , a subsequence (u_{n_{k}}) and a non-increasing sequence of μ-measurable subsets E_{p} with μ(E_{p}) → 0 such that for every p, u_{n_{k}} converges weakly to u in L^{1}(Ω \ E_{p}, μ; IR^{d}). We will say that u is the biting limit of (u_{n_{k}}), which we

denote u_{n_{k}} →^{b} u.

Remark 2.8. If μ(Ω) = ∞, the same result holds with weak convergence in L_{loc}^{1}(Ω \

E_{p}, μ; IR^{d}). We will use the same notation (u_{n_{k}} →^{b} u) in that case.

As a corollary of Theorem 2.5, we obtain:

Theorem 2.9. Let (u_{n_{k}}), ν_{x}, ν_{x}^{∞} and m be as in Theorem 2.5. Then:

(i) f(., u_{n_{k}}) →^{b} ⟨ν_{x}, f⟩ for every Caratheodory function f which satisfies (2.4). (ii) If moreover f belongs to F(Ω × IR^{d}), then f(., u_{n_{k}}) is weakly convergent in L_{loc}^{1}(Ω, μ)

if and only if ⟨ν_{x}^{∞}, |f^{∞}|⟩ = 0 m a.e on Ω. In particular, (u_{n_{k}}) is weakly convergent in L_{loc}^{1}(Ω, μ; IR^{d}) if and only if the measure m vanishes.

Remark 2.10. a) The representation of the biting limit of f(., (u_{n_{k}})) in the form (i) has been already given in [5] (see also [3]) and the associated probability measure ν_{x} can

be seen as a “biting Young measure”. By taking f(x, z) := \frac{d(x, z)}{1 + d(x, z)} where d(x,z) is

the distance of z to some closed valued μ-measurable multifunction F(x), we recover that the probability measure ν_{x} is μ a.e. supported by F(x) whenever dist(u_{n_{k}}(x), F (x)) → 0 μ a.e. (see Remark 2.6). The particular case F (x) = {u(x)} leads to the well-known characterization of μ-convergence in measure:

u_{n_{k}} → u μ-locally in measure ⟺ ν_{x} = δ_{u(x)} μ a.e.  (2.7)

b) The condition m = 0 characterizes the weak compactness in L_{μ}^{1} of the sequence (u_{n}) if moreover (u_{n} μ) is tight.

The next statement gives a geometrical criterium for weak convergence in L^{1}(Ω, μ; IR^{d}) which has been already given by S. Müller [12] in the special case d = 1, μ = Lebesgue measure and F = [0, ∞].

Corollary 2.11. Let F be a closed line free convex subset of IR^{d} and u_{n}, u ∈ L^{1}(Ω, μ; IR^{d})

such that: u_{n} ∈ F μ a.e. , u_{n} μ ⇀^{∗} u μ , u_{n} →^{b} u .

Then u_{n} converges weakly to u in L_{loc}^{1}(Ω, μ; IR^{d}).

Proof. By Theorem 2.5 and Theorem 2.9, one has: u(x) = [ν_{x}] μ a.e. , supp ν_{x}^{∞} ⊂ F^{∞} ∩ S^{d−1} , [ν_{x}^{∞}] = 0 m a.e. . Since F is closed and convex, we have that, for every z_{0} ∈ F, F^{∞} = {z ∈ IR^{d}; z_{0} + tz ∈ F , ∀t ≥ 0}. Hence F^{∞} is also line free and 0 is an extreme point F^{∞}. Then the condition [ν_{x}^{∞}] = 0 implies that the probability measure ν_{x}^{∞} (supported by F^{∞}) is the Dirac mass at 0. Of course, as ν_{x}^{∞} is also supported by S^{d−1}, this is impossible unless m = 0. We conclude by using the last part in assertion (ii) of Theorem 2.9.

3. Examples

To identify the measures m, ν_{x} and ν_{x}^{∞} given by Theorem 2.5, it is enough to test the weak ∗ convergence of f(x, u_{n}(x)) μ for integrands of the class F(Ω × IR^{d}) which do not depend on x. Recall that for such an integrand f, we have (up to a subsequence):

f(u_{n}(x)) μ ⇀^{∗} ⟨ν_{x}, f⟩ μ + ⟨ν_{x}^{∞}, f^{∞}⟩ m  (3.1)

To identify ν_{x}, we take f bounded so that (f(u_{n})) is weakly convergent in L_{loc}^{1} and (3.1) becomes:

f(u_{n}(x)) ⇀ ⟨ν_{x}, f⟩ weakly in L_{loc}^{1}(Ω, μ) for every f ∈ C_{b}(IR^{d})  (3.2)

In a second step, we may identify m and ν_{x}^{∞} by substituting f(z) = |z|φ(\frac{z}{|z|}) in (3.1)

where φ is any continuous function on S^{d−1}:

|u_{n}|φ(\frac{u_{n}}{|u_{n}|}) μ ⇀^{∗} ⟨ν_{x}, |z|φ(\frac{z}{|z|})⟩ μ + ⟨ν_{x}^{∞}, φ⟩ m  (3.3)

In all examples below, μ is the Lebesgue measure on Ω = ]−1, 1[, (u_{n}) is a bounded sequence in L^{1}(Ω) which do not satisfy the uniform integrability property.

Example 3.1. Let u_{n}(x) := −n 1_{]− \frac{1}{n} ,0[} + n 1_{]0, \frac{1}{n} [}. This sequence is bounded in L^{1} and converges ∗-weakly and μ a.e. to 0. However a concentration effect appears at 0 since in that case we have:

ν_{x} = δ_{0} , m = 2δ_{0} , ν_{x}^{∞} = \frac{1}{2} δ_{1} + \frac{1}{2} δ_{−1}

(indeed u_{n}^{+} ⇀^{∗} δ_{0} and u_{n}^{−} ⇀^{∗} δ_{0})

Example 3.2. Here we take the sequence already considered in [3]:

u_{n} = ∑_{k=0}^{n−1} n 1_{[ \frac{k}{n} , \frac{k}{n} + \frac{1}{n^{2}} ]}

(u_{n}) converges to 0 in measure and in the biting sense in L^{1}. Nevertheless as pointed out in [3] , (u_{n}μ) converges ∗-weakly to 1_{[0,1]} μ (where μ is the Lebesgue measure on Ω). In this case the lack of uniform integrability of the sequence is diffused on the whole interval [0, 1] since we find:

ν_{x} = δ_{0} m = 1_{[0,1]} μ ν_{x}^{∞} = δ_{1}

Proof. The convergence of u_{n} to 0 in measure implies that for every f ∈ C_{b}(Ω), one has f(u_{n}) ⇀ f(0) in L_{μ}^{1}. Hence ν_{x} = δ_{0}. Here S^{d−1} = {−1, 1} and as u_{n} ≥ 0, ν_{x}^{∞} is a Dirac mass at z = 1 and m is by (3.3) the weak*-limit of |u_{n}|μ (take φ(z) := |z|). An easy computation gives m = 1_{[0,1]} μ.

Example 3.3. Let: u_{n}(x) = ∑_{k=0}^{n−1} n 1_{[ \frac{k}{n} , \frac{k}{n} + \frac{1}{n^{2}} ]} (cos 2πn^{2}x, sin 2πn^{2}x)

As in Example 3.2, the singularities of u_{n} are uniformly distributed on [0, 1], but we get that ν_{x}^{∞} is uniform on S^{1}:

ν_{x} = δ_{0} m = 1_{[0,1]} μ ν_{x}^{∞} = \frac{1}{2π} H^{1} S^{1}

Proof. For ν_{x} and m, the computation is the same as above. To determine ν_{x}^{∞}, we parametrize S^{1} by θ ∈ [0, 2π] and apply (3.3) with a function φ(θ) where φ is continuous 2π-periodic. We get:

[ ∑_{k=0}^{n−1} n 1_{[ \frac{k}{n} , \frac{k}{n} + \frac{1}{n^{2}} ]} φ(2πn^{2}x)] μ ⇀^{∗} ⟨ν_{x}^{∞}, φ⟩ m

Applying this convergence with a continuous test function Φ and using the mean value theorem for Riemann integrals, it follows:

∫_{Ω}⟨ν_{x}^{∞}, φ⟩Φ(x)m(dx) = lim_{n→∞} ∑_{k=0}^{n−1} n ∫ \frac{k}{n} + \frac{1}{n^{2}}_{\frac{k}{n}} Φ(x)φ(2πn^{2}x)dx

= lim_{n→∞} \frac{1}{n} ∑_{k=0}^{n−1} Φ(\frac{k}{n} + \frac{θ_{k,n}}{n^{2}} )n^{2}∫ \frac{k}{n} + \frac{1}{n^{2}}_{\frac{k}{n}} φ(2πn^{2}x)dx (θ_{k,n} ∈ [0, 1])

= ∫_{0}^{1} Φ(t)dt ∫_{0}^{1} φ(2πu)du .

Hence m is the Lebesgue measure on [0, 1] and ν_{x}^{∞} is uniform on the torus.

4. Proofs

To prove Theorem 2.5, we need first:

Proposition 4.1. Let (u_{n}) be a bounded sequence in L^{1}(Ω, μ; IR^{d}) and F a closed subset of IR^{d} such that u_{n}(x) ϵ F μ a.e.. Then there exist a subsequence (u_{n_{k}}), a nonnegative Radon measure m̃ on Ω and a parametrized family (ν̃_{x})_{x∈Ω} in P(Ω, m̃; B̄_{d}) such that:

(f(x, u_{n_{k}}(x)) μ ⇀^{∗} (∫B̄_{d} f̃(x, z)ν̃_{x}(dz)) m̃  (4.1)

for every f in F(Ω × IR^{d}). Moreover, let m̃ = p̃μ + m̃_{s} with μ ⊥ m̃_{s} be the Lebesgue-Nikodym decomposition of m̃ with respect to μ and let F̃ be the closed subset of B̄_{d} defined by F̃ = cl\{\frac{z}{1 + |z|}; z ϵ F \}.

Then the following properties hold:

(i) supp m̃ ⊂ supp μ and supp ν̃_{x} ⊂ F̃ for m̃ a.e. x ∈ Ω (ii) p̃(x) = (∫_{B_{d}}(1 − |z|)ν̃_{x}(dz))^{−1} ≥ 1 for μ a.e. x ∈ Ω (iii) ν̃_{x}(S^{d−1}) = 1 for m̃_{s} a.e. x ∈ Ω.

Proof. By (2.4), for every f ∈ F, the sequence f(x, u_{n}) μ is bounded hence ∗-weakly relatively compact in M(Ω). Therefore we need only to check (4.1) on relatively compact subsets of Ω so that, without any loss of generality, we may assume that μ(Ω) < +∞. Then we define a bounded sequence of Radon measures on Ω × B̄_{d} by setting for every φ ∈ C_{0}(Ω × B̄_{d}):

⟨L_{n}, φ⟩ = ∫_{Ω×B̄_{d}} φdL_{n} := ∫_{Ω} φ(x, \frac{u_{n}(x)}{1 + |u_{n}(x)|})(1 + |u_{n}(x)|)dμ

(indeed |⟨L_{n}, φ⟩| ≤ |φ|_{∞}(μ(Ω) + sup ∥u_{n}∥_{L^{1}}). Hence there exists a subsequence (L_{n_{k}}) and a bounded Radon measure L such that for every φ ∈ C_{0}(Ω × B̄_{d}):

lim_{k→∞}⟨L_{n_{k}}, φ⟩ = ⟨L, φ⟩ = ∫_{Ω×B̄_{d}} φ(x, z)L(dxdz)  (4.2)

Clearly by taking φ depending only on x in (4.2), we identify m̃ as the canonical projection of L on Ω that is m̃(E) = L(E × B̄_{d}) for every Borel subset E of Ω. Now by a classical slicing argument for measures (often referred as a disintegration theorem, see for example [13]), there exists a family of probability measures in P(Ω, m̃; B̄_{d}) such that:

⟨L, φ⟩ = ∫_{Ω}(∫B̄_{d} φ(x, z) ν̃_{x}(dz)) m̃(dx)

Let f belong to the class F(Ω × IR^{d}) and take for φ in (4.2) the function defined by φ(x, z) = Φ(x)f̃(x, z) where Φ runs over C_{0}(Ω). We are led to:

∫_{Ω} f(x, u_{n_{k}})Φ(x) dμ → ∫_{Ω}⟨ν̃_{x}, f̃⟩Φ(x) dm̃

which is equivalent to (4.1). Moreover by taking f(z) := 1 + |z| in (4.1) we find that m̃ is the weak* limit of the sequence (1 + |u_{n_{k}}|) μ. Hence m̃ ≥ μ and supp m̃ ⊂ supp μ. To prove (i), we choose a sequence of continuous functions g_{p} : B̄_{d} → [0, 1] such that g_{p} = 1 on F̃ and g_{p} → 0 on B̄_{d} \ F̃. Then applying (4.1) with f(z) := (1 + |z|)g_{p}(\frac{z}{1+|z|}), we obtain that f(u_{n_{k}})μ converges ∗-weakly to ⟨ν̃_{x}, g_{p}⟩m̃. As f(u_{n_{k}}) = (1 + |u_{n_{k}}|) μ a.e., this limit coincides with m̃ which yields: ⟨ν̃_{x}, g_{p}⟩ = 1 m̃ a.e.. By letting p tend to ∞, we obtain ν̃_{x}(F̃) = 1 m̃ a.e. which proves the assertion (i). Let now apply (4.1) with f(x, z) := 1 that is f̃(x, z) = 1 − |z|. We get: μ = ⟨ν̃_{x}, f̃⟩m̃ which according to the Lebesgue-Nikodym decomposition of m̃ splits into:

⟨ν̃_{x}, 1 − |z|)⟩ p̃(x) = 1 μ a.e.  (4.3)
⟨ν̃_{x}, 1 − |z|)⟩ = 0 m̃_{s} a.e.  (4.4)

Since 1−|z| ≤ 1 on B̄_{d}, (4.3) yields (ii) and (4.4) that for m̃_{s} a.e. on Ω , ν̃_{x} is concentrated on {|z| = 1} = S^{d−1} that is (iii). The proof of Proposition 4.1 is achieved.

Proof of Theorem 2.5. Let m̃, p̃, ν̃_{x} defined as in Proposition 4.1. For μ a.e. x ∈ Ω , we define a probability measure on IR^{d} by setting for every bounded continuous φ on IR^{d}:

⟨ν_{x}, φ⟩ := p̃(x) ∫_{\{|z|<1\}}(1 − |z|)φ(\frac{z}{1 − |z|})ν̃_{x}(dz) = p̃(x) ∫_{B} φ̃(z)ν̃_{x}(dz)  (4.5)

(indeed for φ ≡ 1, we get from Proposition 4.1 (ii): ⟨ν_{x}, 1⟩ = p̃(x)⟨ν̃_{x}, 1 − |z|⟩ = 1 μ a.e.). We define also the bounded Radon measure m := ν̃_{x}(S^{d−1}) m̃ . We have supp m ⊂ supp m̃ ⊂ supp μ (see Proposition 4.1 (i)). Moreover the Lebesgue-Nikodym decomposition of m with respect to μ reads as:

m = p̃(x)ν̃_{x}(S^{d−1})μ + m̃_{s} .  (4.6)

Now we define for m a.e. x ∈ Ω a probability measure on S^{d−1} by setting:

⟨ν_{x}^{∞}, φ⟩ := \frac{1}{ν̃_{x}(S^{d−1})} ∫_{S^{d−1}} φ(z)ν̃_{x}(dz) .  (4.7)

Using (4.5)(4.6)(4.7) and noticing that f^{∞}(x, .) = f̃(x, .) on S^{d−1} for every f in F(Ω×IR^{d}) and that ν̃_{x}(S^{d−1}) = 1 m̃_{s} a.e. (see assertion (iii) of Proposition 4.1), one gets:

⟨ν_{x}, f⟩μ + ⟨ν_{x}^{∞}, f^{∞}⟩m
= [⟨ν_{x}, f⟩ + p̃(x)ν̃_{x}(S^{d−1})⟨ν_{x}^{∞}, f^{∞}⟩] μ + ⟨ν_{x}^{∞}, f^{∞}⟩m̃_{s}
= (∫_{B_{d}} f̃(x, z)p̃(x)ν̃_{x}(dz) + ∫_{S^{d−1}} f̃(x, z)p̃(x)ν̃_{x}(dz)) μ

+ (\frac{1}{ν̃_{x}(S^{d−1})} ∫B̄_{d} f̃(x, z)ν̃_{x}(dz))) m̃_{s}

= ∫B̄_{d} f̃(x, z)ν̃_{x}(dz) [p̃(x)μ + m̃_{s}]
= ⟨ν̃_{x}, f̃⟩m̃ .

Hence (4.1) can be reformulated as:

(f(x, u_{n_{k}}(x)) μ ⇀^{∗} ⟨ν_{x}, f⟩μ + ⟨ν_{x}^{∞}, f^{∞}⟩m  (4.8)

for every f in F(Ω × IR^{d}). The first statement of (i) is easily obtained by applying (4.8) to f(x, z) := |z|. By the boundedness of (u_{n}) in L^{1}, we have ∫_{Ω}⟨ν_{x}, |z|⟩dμ < +∞. To prove the second statement of (i), we consider a bounded continuous function φ on IR^{d} such that φ = 0 on the closed subset F . We have to show that ⟨ν_{x}, φ⟩ = 0 μ a.e. on Ω. For such a function φ, the associated φ̃ (see (2.5) and Lemma 2.2 (i)) vanishes on F̃ ∪ S^{d−1}. By assertion (i) of Proposition 4.1, ν̃_{x} is concentrated on F̃ m̃ a.e.( hence μ a.e. since m̃ ≥ μ ) so that using (4.5), we can write:

⟨ν_{x}, φ⟩ = p̃(x) ∫_{F̃ ∩B} φ̃(z)ν̃_{x}(dz) = 0 μ a.e. x ∈ Ω.

Similarly let φ be a continuous function on S^{d−1} satisfying φ = 0 on F^{∞} ∩ S^{d−1}(= F̃ ∩ S^{d−1}). From (4.7), we have:

⟨ν_{x}^{∞}, φ⟩ := \frac{1}{ν̃_{x}(S^{d−1})} ∫_{S^{d−1}∩F̃} φ(z)ν̃_{x}(dz) = 0 m a.e. x ∈ Ω,

wich proves that ν_{x}^{∞} is concentrated on F^{∞} that is (ii). Let us now prove (iii): we consider the sequence (f_{p}) in F given by Lemma 2.3. Apply the first statement of our theorem to f_{p} yields for every p:

lim inf_{k→∞} ∫_{Ω} f(x, u_{n_{k}})dμ ≥ ∫_{Ω}⟨ν_{x}, f_{p}(x, .)⟩ dμ + ∫_{Ω}⟨ν_{x}^{∞}, f_{p}^{∞}(x, .)⟩ dm  (4.9)

Set: g_{p}(x) := ⟨ν_{x}, f_{p}(x, .)⟩, h_{p}(x) := ⟨ν_{x}^{∞}, f_{p}^{∞}(x, .)⟩. As g_{p} and h_{p} are non decreasing and have a uniform lower bound in L_{μ}^{1} (resp. in L_{m}^{1}): g_{p}(x) ≥ −Λ(1 + ⟨ν_{x}, |z|⟩ (resp. h_{p}(x) ≥ −Λ ), we conclude by letting p tend to ∞ in (4.9) using monotone convergence theorem. The proof of Theorem 2.5 is finished.

Proof of Theorem 2.9. By a trivial localization argument, we can assume that the sequence (u_{n}μ) is tight and that μ(Ω) is finite (see the Remark 2.8). As (v_{k}) is bounded in L_{μ}^{1} where v_{k}(x) := f(x, u_{n_{k}}(x)), by Lemma 2.7 we can assume, possibly passing to a subsequence, that it converges in the biting sense to some limit l(x). If we prove that l(x) ≡ ⟨ν_{x}, f⟩ , then by unicity of the limit, the whole sequence f(x, u_{n_{k}}(x)) will satisfy the assertion (i).

First step: we assume that (v_{k}) is uniformly integrable so that it converges weakly to l(x) in L_{μ}^{1}. Assume first that f is bounded and let us show that l(x) = ⟨ν_{x}, f(x, .)⟩ μ a.e. on every compact subset K of Ω. By the theorem of Scorzá-Dragoni (see for example [14]), for every ε > 0, there exists a compact K_{ε} ⊂ K and a continuous g_{ε} on Ω × IR^{d} such that:

μ(K \ K_{ε}) < ε g_{ε} ≡ f on K_{ε} × IR^{d} .  (4.10)

By truncating, we can assume that g_{ε} is bounded so that g_{ε} belongs to F(Ω × IR^{d}). The weak limit of g_{ε}(., u_{n_{k}}) given by Theorem 2.5 is μ-absolutely continuous and coincides with l on K_{ε}, hence by (4.10):

l(x) = ⟨ν_{x}, g_{ε}(x, .)⟩ = ⟨ν_{x}, f(x, .)⟩ μ a.e. on K_{ε}.

The equality μ a.e. on Ω follows by letting ε tend to 0. To relax the boundedness assumption on f, we approximate f by the sequence (f_{p}) defined by f_{p}(x, z) :=

sup{−p, inf{f(x, z), p)} and obtain as above the weak convergence of the sequence v_{k}^{(p)} := f_{p}(., u_{n_{k}}). More precisely:

v_{k}^{(p)} ⇀ l_{p} weakly in L_{μ}^{1} as k → ∞ ,  l_{p}(x) = ⟨ν_{x}, f_{p}(x, .)⟩.  (4.11)

By the uniform integrability of (v_{k}), by (4.11) and the weak lower semicontinuity of the L^{1}-norm, we are led to:

lim sup_{p→∞} ||l_{p} − l||_{L^{1}} ≤ lim sup_{p→∞} lim inf_{k→∞} ||v_{k}^{(p)} − v_{k}||_{L^{1}} = 0

which proves that l is the strong limit of l_{p} in L_{μ}^{1} . By the growth condition (2.5) and the assertion (i) of Theorem 2.5, we can apply dominated convergence theorem to the integral ∫ f_{p}(x, z)ν_{x}(dz) to see that l_{p} converges also μ a.e. to l = ∫ f(x, z)ν_{x}(dz).

Second step: Now we consider the general case. By Lemma 2.7, there exists a nonincreasing sequence (E_{p}) such that μ(E_{p}) → 0 and v_{k} converges weakly to l in L^{1}(Ω\E_{p}, μ; IR^{d}). For every p, we set A_{p} = Ω\E_{p} and consider the Caratheodory function g_{p}(x, z) := 1_{A_{p}}(x)f(x, z) which satisfies (2.4). Noticing that v_{k}1_{A_{p}} := g_{p}(., u_{n_{k}}), we can apply step 1 so that: v_{k}1_{A_{p}} ⇀ ⟨ν_{x}, g_{p}(x, .)⟩ weakly in L_{μ}^{1}. Then by the unicity of the weak limit of (v_{k}) on every set A_{p}, we get l(x) = ⟨ν_{x}, f(x, .)⟩ μ a.e. on A_{p}. The conclusion follows by letting p tend to ∞.

Let us now prove (ii): by the Dunford-Pettis theorem, the weak compactness in L_{μ}^{1} of

v_{k} := f(x, u_{n_{k}}) holds if and only if η(R) := lim sup_{k→∞} ∫_{|v_{k}|≥R} |v_{k}|dμ tends to 0 as R → ∞.

Let us define:

f_{R}(x, z) := R h (\frac{|f(x, z)|}{R})

where

h(t) = 0 if 0 ≤t< \frac{1}{2} , h(t) = 2t − 1 if \frac{1}{2} ≤t ≤ 1 , h(t) = t if t > 1 .

It is easy to check that f_{R} belongs to the class F(Ω × IR^{d}) and that:

f_{2R}(., u_{nk}) ≤ |v_{k}|1_{{|v_{k}|>R}} ≤ f_{R}(., u_{n_{k}}) and f_{R}^{∞} = f_{2R}^{∞} = |f|^{∞} .

Passing to the limit as k → ∞ and using Theorem 2.5, we get for every R:

\begin{aligned}∫_{Ω}⟨ν_{x}, f_{2R}⟩dμ + & ∫_{Ω}⟨ν_{x}^{∞}, |f^{∞}|⟩dm \\ & ≤ η(R) ≤ ∫_{Ω}⟨ν_{x}, f_{R}⟩dμ + ∫_{Ω}⟨ν_{x}^{∞}, |f^{∞}|⟩dm \end{aligned}  (4.12)

Then letting R tend to ∞ in (4.12) and using dominated convergence theorem:

η_{∞} := lim_{R→∞} η(R) = ∫_{Ω}⟨ν_{x}^{∞}, |f^{∞}|⟩dm  (4.13)

The first equivalence in the assertion (ii) of Theorem 2.9 follows. In the particular case f(x, z) = |z|, (4.13) reduces to η_{∞} = m(Ω). Hence the weak compactness of (u_{n_{k}}) in L_{μ}^{1} is equivalent to m = 0 .

Remark 4.2. The real η_{∞} in (4.13) is the modulus of uniform integrability associated with the sequence (v_{k}) intoduced by H.P.Rosenthal. Note also that in the proof of the assertion (i), the measurable dependence in x is overcomed as in [14] by a Lusin’s type argument which is very far from the technique used in [1,2,3].

5. Applications in Calculus of Variations

In connection with lower semicontinuity results of integral functionals with respect to the weak∗- convergence of measures, we are looking for an optimal lower bound for lim inf_{n→∞} ∫ f(x, u_{n})dμ when (u_{n}) is a sequence in L^{1}(Ω, μ; IR^{d}) converging ∗-weakly to some measure λ ∈ M(Ω; IR^{d}) . The lower bound we propose hereafter takes into account the possible gap between the biting limit of u_{n} and the μ-absolutely continuous part of the limit λ.

Theorem 5.1. Let f(x, z) : Ω×IR^{d} → (−∞, ∞] be a lower semicontinuous integrand, convex with respect to z such that f(x, 0) is locally bounded on Ω and verifying:

∃C > 0 , ∃a ∈ L_{μ}^{1}(Ω) :  f(x, z) ≥ −C|z| + a(x)  (5.1)

Let λ ∈ M(Ω; IR^{d}) with Lebesgue-Nikodym decomposition λ = u^{a}μ + λ_{s}. Then for every sequence (u_{n}) such that:

u_{n} μ ⇀^{∗} λ  in M(Ω; IR^{d})  ,  u_{n} →^{b} u^{b}

one has:

lim inf_{n→∞} ∫_{Ω} f(x, u_{n})dμ ≥ ∫_{Ω} f(x, u^{b})dμ + ∫_{Ω} f^{∞}(x, u^{a} − u^{b})dμ + ∫_{Ω} f^{∞}(x, \frac{dλ_{s}}{d|λ_{s}|})d|λ_{s}|

Notation: When h(x,z) is a Borel function on Ω × IR^{d} positively 1-homogeneous with respect to z and satisfy the lower bound h(x, z) > −C(1+|z|) the integral ∫_{B} h(x, \frac{dλ}{d|λ|})d|λ|

is well defined for any measure in M(Ω; IR^{d}) and any Borel subset B. As the expression ∫ h(x, \frac{dλ}{dθ})dθ does not depend on the positive measure θ provided λ ≪ θ, this integral will be denoted simply ∫_{B} h(x, λ) and the corresponding Borel measure by h(x, λ) (see [8]).

Remark 5.2. a) From Theorem 5.1 and assertion (i) of Lemma 5.4 below, we recover the lower bound inequality obtained by Goffmann&Serrin [8]:

lim inf_{n→∞} ∫_{Ω} f(x, u_{n})dμ ≥ ∫_{Ω} f(x, u^{a})dμ + ∫_{Ω} f^{∞}(x, λ_{s})  (5.2)

Let us notice that this inequality is not optimal: let us consider the sequence (u_{n}) of Example 3.2 and the integrand f(x, z) := 1 + |z| . We find:

lim_{n→∞} ∫_{Ω} f(x, u_{n}(x))dμ = ∫_{Ω} f(x, u^{b})dμ + ∫_{Ω} f^{∞}(x, u^{a} − u^{b})dμ + ∫_{Ω} f^{∞}(x, λ_{s}) = 3

> ∫_{Ω} f(x, u^{a})dμ + ∫_{Ω} f^{∞}(x, λ_{s}) = 2

b) It is possible to extend Theorem 5.1 by relaxing the lower semicontinuity assumption of f with respect to x. In that case, the integrand h_{f} has to be replaced by a ”μ-essentiel regularization already used in [10] (see Proposition 5.5 below).

In the following theorem, we obtain a strong convergence result under strict convexity assumption:

Theorem 5.3. Let f(x, z) : Ω×IR^{d} → (−∞, ∞] be a lower semicontinuous integrand, strictly convex with respect to z, such that (5.1) holds and f(., 0) is locally bounded on Ω. Let (u_{n}) be a sequence in L^{1}(Ω, μ; IR^{d}) such that:

u_{n} μ ⇀^{∗} λ in M(Ω; IR^{d})  ,  λ = u μ + λ_{s} with λ_{s}⊥μ ,

lim sup_{n→∞} ∫_{Ω} f(x, u_{n})dμ ≤ ∫_{Ω} f(x, u)dμ + ∫_{Ω} f^{∞}(x, λ_{s}) < +∞ .

Then the following properties hold:

(i) u_{n} converges to u μ-locally in measure (and u_{n} →^{b} u). (ii) u_{n} → u strongly in L_{loc}^{1}(Ω, μ; IR^{d}) if moreover λ_{s} = 0. (iii) Assume that f^{∞}(x, .) is strictly convex for all x ∈ Ω, then

g(x, u_{n})μ ⇀^{∗} g(x, u) μ + g^{∞}(x, λ_{s}) in M(Ω; IR^{d}),

for every g in the class F(Ω × IR^{d}).

Moreover in (ii) the convergence is strong in L^{1}(Ω, μ; IR^{d}) and in (iii) the convergence is tight in M(Ω; IR^{d}) if f satisfies: ∃C > 0 ∃b ∈ L_{μ}^{1} f(x, z) ≥ C|z| − b(x) .

Remark 5.4. a) Recall that for a convex 1-homogeneous function h : IR^{d} → (−∞, +∞] strict convexity means that h(y + z) = h(y) + h(z) with |y|=|z|=1 implies y = z. Note that the strict convexity of a function g on IR^{d} does not imply the strict convexity of the 1-homogeneous function g^{∞} (Ex. Let g : (z_{1}, z_{2}) ∈ IR^{2} → (1 + |z_{1}|^{2})^{1/2} + (1 + |z_{2}|^{2})^{1/2}, then g^{∞}(z_{1}, z_{2}) = |z_{1}| + |z_{2}|).

b) The property (ii) has been proved in [11] in case f is independent of x. The property (iii) appears already in [9] in case of 1-homogeneous integrands.

To prove Theorems 5.1 and 5.3, we will use some facts of convex analysis collected in Lemma 5.5 below (proved in Appendix):

Lemma 5.5. Let g : IR^{d} → (−∞, +∞] be a convex proper l.s.c. function and ν a probability on IR^{d} such that ∫_{IR}_{d} |z|ν(dz) < +∞. Then:

(i) g(y + z) ≤ g(y) + g^{∞}(z) for all y, z ∈ IR^{d}. g(y + z) = g(y) + g^{∞}(z) < +∞ ⇒ z = 0 whenever g is strictly convex.

(ii) g([ν]) ≤ ⟨ν, g⟩ (Jensen inequality) g([ν]) = ⟨ν, g⟩ < +∞ ⇒ ν is a Dirac mass

whenever g is strictly convex. (iii) ⟨ν, g^{∞}⟩ > 0 whenever g is strictly convex, [ν] = 0 and ν is not the Dirac mass at 0

(observe that g^{∞} is not assumed to be strictly convex).

Proof of Theorem 5.1. Applying Theorem 2.5 to the sequence (u_{n}), we can associate a subsequence (u_{n_{k}}), a measure m and parametrized probabilities ν_{x} and ν_{x}^{∞} such that all conclusions of this theorem are fulfilled and such that: lim inf_{n→∞} ∫_{Ω} f(x, u_{n})dμ =

lim_{k→∞} ∫_{Ω} f(x, u_{n_{k}})dμ. As u_{n} μ ⇀^{∗} λ, we find that: λ = [ν_{x}]μ + [ν_{x}^{∞}]m. By Theorem 2.9, we have also: u^{b} = [ν_{x}] . Hence:

[ν_{x}^{∞}]m = λ − u^{b} μ = (u^{a} − u^{b})μ + λ_{s}.  (5.3)

By the assumptions on f (see the Remark 2.4), we have h_{f} = f^{∞} so that by the assertion (iii) of Theorem 2.5, Jensen inequality and (5.3), we conclude:

lim inf_{n→∞} ∫_{Ω} f(x, u_{n})dμ = lim_{k→∞} ∫_{Ω} f(x, u_{n_{k}})dμ
≥ ∫_{Ω}⟨ν_{x}, f(x, .)⟩dμ + ∫_{Ω}⟨ν_{x}^{∞}, f^{∞}(x, .)⟩dm

≥ ∫_{Ω} f(x, [ν_{x}]) dμ + ∫_{Ω} f^{∞}(x, [ν_{x}^{∞}])dm  (5.4)
= ∫_{Ω} f(x, u^{b}) dμ + ∫_{Ω} f^{∞}(x, (u^{a} − u^{b})μ + λ_{s})
= ∫_{Ω} f(x, u^{b}) dμ + ∫_{Ω} f^{∞}(x, (u^{a} − u^{b}))dμ + ∫_{Ω} f^{∞}(x, λ_{s}) .

Proof of Theorem 5.3. Take (u_{n_{k}}), m, ν_{x} and ν_{x}^{∞} as in the proof of Theorem 5.1 and set:

\begin{aligned}I_{1} & := ∫_{Ω} f(x, u)dμ + ∫_{Ω} f^{∞}(x, λ_{s}) \\ I_{2} & := ∫_{Ω} f(x, u^{b}) dμ + ∫_{Ω} f^{∞}(x, (u − u^{b}))dμ + ∫_{Ω} f^{∞}(x, λ_{s}) (u^{b} = [ν_{x}]) \\ I_{3} & := ∫_{Ω} f(x, [ν_{x}]) dμ + ∫_{Ω} f^{∞}(x, [ν_{x}^{∞}]) dm \end{aligned}

By (5.4), we have: lim inf_{n→∞} ∫_{Ω} f(x, u_{n})dμ ≥ I_{3} ≥ I_{2}. By assumption and by Lemma 5.5 (i): I_{2} ≥ I_{1} ≥ lim sup_{n→∞} ∫_{Ω} f(x, u_{n})dμ. It follows: I_{1} = I_{2} = I_{3}. Since I_{2} = I_{1} < +∞, we deduce from (i) of Lemma 5.5 and (5.3) that:

u = u^{b} μ a.e. , λ_{s} = [ν_{x}^{∞}] m  (5.5)

Now since I_{2} = I_{3} , by the assertion (ii) of Lemma 5.5 we get:

⟨ν_{x}, f(x, .)⟩ = f(x, [ν_{x}]) μ a.e.  (5.6)
⟨ν_{x}^{∞}, f^{∞}(x, .)⟩ = f^{∞}(x, [ν_{x}^{∞}]) m a.e.  (5.7)

The strict convexity of f(x, .) , (5.6) and (5.5) imply that ν_{x} is μ a.e. concentrated at u(x) which proves the assertion (i) of our theorem (see (2.7)). Let us prove (ii): by Vitali’s theorem and the μ-convergence in measure of u_{n} already proved in (i), it is enough to show that (u_{n}) is locally uniformly integrable which from Theorem 2.9 (ii) is equivalent to say that m = 0. Assume that m does not vanish. Then by (5.5) and (5.7), the assumption λ_{s} = 0 yields: [ν_{x}^{∞}] = 0 and < ν_{x}^{∞}, f^{∞}(x, .) >= 0 m a.e.. This is incompatible with the assertion (iii) of Lemma 5.5. Assume now that for every x, the function f^{∞}(x, .) is strictly convex. Then by (5.7) and by Lemma 5.5 (ii)), ν_{x}^{∞} is for a.e. x a Dirac mass concentrated at [ν_{x}^{∞}] which is by (5.5) nothing else but the Radon Nikodym derivative of λ_{s} with respect to m. Let g a function in the class F(Ω × IR^{d}). The sequence g(x, u_{n}(x)) μ is bounded hence weak ∗-relatively compact in M(Ω; IR^{d}). Consider a weakly convergent subsequence. Noticing that all previous arguments appply to the corresponding subsequence (u_{n_{k}}) of the original sequence (u_{n}), we can identify the limit through Theorem 2.5:

\begin{aligned}g(x, u_{n_{k}}) μ ⇀^{∗} & ⟨ν_{x}, g(x, .)⟩μ + ⟨ν_{x}^{∞}, g^{∞}(x, .)⟩m \\ & = g(x, u(x)) μ + g^{∞}(x, \frac{dλ_{s}}{dm}) m . \end{aligned}

By the usual unicity argument, the whole sequence g(x, u_{n})μ converges ∗-weakly. The assertion (iii) is proved. To complete the proof we have only to show that the sequence α_{n} := f(x, u_{n}) μ is tight. Indeed under the coercivity condition on f, this will imply the tightness of |u_{n}|μ and then that of the sequence g(x, u_{n})μ for all g in F(Ω×IR^{d}) (use the growth condition (2.4)). Let α := f(x, u) μ + f^{∞}(x, λ_{s}) . By localizing the semicontinuity result of Theorem 5.1 and the inequality (5.2), we have for every open subset ω of Ω: lim inf_{n→∞} α_{n}(ω) ≥ α(ω). By assumption, we also have: lim sup_{n→∞} α_{n}(Ω) ≤ α(Ω) < +∞. Since by (5.1) the negative part of α_{n} is uniformly bounded in variation, the conclusion follows.

Finally we indicate breefly how to relax in Theorem 5.1 the lower semicontinuity assumption of f with respect to x: let f(x, z) be a convex normal integrand (that is measurable in (x, z), convex and l.s.c. in z). Assume that:

∃φ_{0} ∈ C_{0}(Ω; IR^{d}) , ∃a ∈ L_{μ}^{1} : f(x, z) ≥ φ_{0}(x).z − a(x) .  (5.8)

Let us define for every (x, z) in Ω × IR^{d}:

h_{f}^{μ}(x, z) := sup {φ(x).z | φ ∈ C_{0}(Ω; IR^{d}) , ∫_{Ω} f^{⋆}(x, φ(x))μ(dx) < ∞ } (5.9)

where f^{⋆}(x, z^{⋆}) := sup_{z} {z^{⋆}.z−f(x, z)}. The function h_{f}^{μ}(x, z) is l.s.c.,convex and positively

1-homogeneous in z and we have (see [10], Prop. 7):

\{ \begin{matrix}h_{f}^{μ}(x, .) ≤ f^{∞}(x, .) & for μ a.e. x ∈ Ω \\ h_{f}^{μ} ≥ f^{∞} & if the multifunction x → epif^{⋆}(x, .) is l.s.c. \end{matrix}  (5.10)

In particular h_{f}^{μ} ≥ f^{∞} holds if f satisfies the conditions of Theorem 5.1 and h_{f}^{μ}(x, .) ≡

h_{f}(x, .) ≡ f^{∞}(x, .) for every x ∈ supp μ if moreover f belongs to F(Ω × IR^{d}) (see [10], Theorem 8).

Proposition 5.6. Let f(x, z) : Ω × IR^{d} → (−∞, ∞] be a convex normal integrand satisfying (5.8). Let λ ∈ M(Ω; IR^{d}) with Lebesgue-Nikodym decomposition λ = u^{a}μ + λ_{s}. Then for every sequence (u_{n}) such that:

u_{n}μ ⇀^{∗} λ in M(Ω; IR^{d})  ,  u_{n} →^{b} u^{b}

one has:

lim inf_{n→∞} ∫_{Ω} f(x, u_{n})dμ ≥ ∫_{Ω} f(x, u^{b})dμ + ∫_{Ω} h_{f}^{μ}(x, u^{a}−u^{b})dμ + ∫_{Ω} h_{f}^{μ}(x, λ_{s})

Remark 5.7. By eliminating u^{b} in the inequality above, we derive an inequality similar to (5.2):

lim inf_{n→∞} ∫_{Ω} f(x, u_{n})dμ ≥ ∫_{Ω} g(x, u^{a})dμ + ∫_{Ω} h_{f}^{μ}(x, λ_{s})

J.J.Alibert, G. Bouchitté / Non-uniform integrability and generalized young measures 145

where the integrand g is defined for every (x, z) ∈ Ω × IR^{d} by: g(x, z) := inf{f(x, z′) + h_{f}^{μ}(x, z − z′); z′ ∈ IR^{d}}. This lower bound has been already obtained in [10].

Proof of Proposition 5.5. As done before we can reduce easily to the case μ(Ω) < +∞. Let (E_{p}) a nonincreasing sequence such that: μ(E_{p}) → 0 and u_{n} ⇀ u^{b} in L_{μ}^{1}(Ω\E_{p}) (see Lemma 2.7). By a classical lower semicontinuity result, we have:

lim inf_{n→∞} ∫_{Ω\E_{p}} f(x, u_{n}(x))dμ ≥ ∫_{Ω\E_{p}} f(x, u^{b}(x))dμ .  (5.11)

On the other hand by Fenchel’s inequality we have for every φ in C_{0}(Ω × IR^{d}):

∫_{E_{p}} f(x, u_{n}(x))dμ ≥ ∫_{Ω} φ(x).u_{n}(x)1_{E_{p}}dμ − ∫_{E_{p}} f^{⋆}(x, φ(x))dμ

Since u_{n}1_{Ω\E_{p}} μ ⇀^{∗} u^{b} 1_{Ω\E_{p}} μ we can write: u_{n}1_{E_{p}}μ ⇀^{∗} (u^{a} −u^{b})μ+u^{b}1_{E_{p}}μ+λ_{s}

so that:

lim inf_{n→∞} ∫_{E_{p}} f(x, u_{n}(x))dμ ≥

∫_{Ω} φ(x).(u^{a} − u^{b})dμ + ∫_{Ω} φ(x).dλ_{s} + ∫_{E_{p}}(φ(x).u^{b}(x) − f^{⋆}(x, φ(x)))dμ .

(5.12)

Collecting (5.11)(5.12) and passing to the limit as p → ∞, we obtain using dominated convergence theorem:

lim inf_{n→∞} ∫_{Ω} f(x, u_{n}(x))dμ ≥ ∫_{Ω} f(x, u^{b})dμ + ∫_{Ω} φ(x).(u^{a}−u^{b})dμ + ∫_{Ω} φ(x).dλ_{s} (5.13)

for every φ ∈ C_{0}(Ω; IR^{d}) such that ∫_{Ω} f^{⋆}(x, φ(x))dμ < ∞. Then we take the supremum in (5.13) with respect to all functions φ satisfying the condition above and conclude by applying a result of commutativity of ∫ and sup (see [10], Theorem 1).

Appendix

Proof of Lemma 2.2. Under (i), f^{∞} := 0 and the continuity of f̃ is trivial. Under the homogeneity assumption (ii), f̃ is the restriction of f to Ω × B̄_{d}. Assume now (iii) and let (x_{n}, z_{n}) a sequence converging to (x, z) in Ω × B̄_{d}. We have to show that f̃(x_{n}, z_{n}) → f̃(x, z). If |z| < 1, it comes down to the continuity of f. In case |z| = 1 ,we use the convexity of f(x, .)( which guarantees the existence of f^{∞}). Moreover for every t > 0, we have eventually t > 1 −|z_{n}|, hence:

tf(x_{n}, \frac{z_{n}}{t}) + (1 − |z_{n}| − t)f(x_{n}, 0) ≤ f̃(x_{n}, z_{n}) ≤ f^{∞}(x_{n}, z_{n}) + (1 − |z_{n}|)f(x_{n}, 0) .

We conclude by passing to the limit as n → ∞ (using the continuity of f and f^{∞}) and then as t → 0.

Proof of Lemma 2.3. Let g be defined on Ω×B_{d} by g(x, z) := (1−|z|)f(x, \frac{z}{1 − |z|})

and denote by \overline{g} its lower semicontinuous extension on Ω × \overline{B}_{d}. As f is l.s.c. and satisfies (5.1), we have \overline{g} = g on Ω × B_{d} and \overline{g} is lower bounded by −Λ. Moreover we have: \overline{g} = h_{f} on Ω × S^{d−1}. Indeed let (x, z) ∈ Ω × S^{d−1} and (x_{k}, z_{k}, t_{k}) a sequence such that x_{k} → x, z_{k} → z, t_{k} → +∞. Then:

lim inf_{k} \frac{f(x_{k}, t_{k}z_{k})}{t_{k}} = lim inf_{k}(\frac{1}{t_{k}} + |z_{k}|)g(x_{k}, \frac{z_{k}}{\frac{1}{t_{k}} + |z_{k}|}) ≥ \overline{g}(x, z)

Hence h_{f}(x, z) ≥ \overline{g}(x, z). Conversely choosing the particular sequence t_{k} = \frac{1}{1−|z_{k}|}, we get:

lim inf_{k} g(x_{k}, z_{k}) = lim inf_{k} \frac{1}{t_{k}} f(x_{k}, t_{k}z_{k}) ≥ h_{f}(x, z) (hence \overline{g}(x, z) ≥ h_{f}(x, z)). We can construct a non decreasing sequence (g_{p}) in C_{b}(Ω × B̄_{d}) such that g_{p} ≥ −Λ and

sup_{p} g_{p} = \overline{g} . Then the sequence (f_{p}) defined by f_{p}(x, z) := (1+|z|)g_{p}(x, \frac{z}{1 + |z|}) satisfies

all conditions of the lemma ( note that f_{p}^{∞}(x, z) = g_{p}(x, z) on Ω × S^{d−1}) . Let us prove finally that h_{f} ≡ f^{∞} when f(x, .) is convex and f(x, 0) locally upper bounded (in order to justify the remark after the lemma). Let (x_{k}, z_{k}) → (x, z) and t_{k} → ∞. Then for every t > 0, we have using the convexity assumption, the lower semicontinuity of f and the upper bound (2.4):

lim inf_{k} \frac{f(x_{k}, t_{k}z_{k})}{t_{k}} ≥ lim inf_{k} [\frac{f(x_{k}, tz_{k})}{t} − (\frac{1}{t} − \frac{1}{t_{k}})f(x_{k}, 0)] ≥ \frac{f(x, tz)}{t} − \frac{Λ}{t} .

The conclusion follows by letting t → ∞.

Proof of Lemma 5.4. Let y, z ∈ IR^{d} such that g(y) < +∞ and z ≠ 0. By the convexity of g, the function h(t) := \frac{g(y+tz)−g(y)}{t} is nondecreasing on (0, +∞) and converges to g^{∞}(z) as t → ∞. Hence g^{∞}(z) ≥ h(1). Assuming moreover g strictly convex and g^{∞}(z) finite yields g^{∞}(z) > h(1). The assertion (i) is proved. The assertion (ii) is classical (see for example [13, p.273]). To prove (iii), let us assume that ⟨ν, g^{∞}⟩ = 0. Using (i) and applying (ii) with g_{y}(z) := g(y + z) (g(y) < +∞), we get:

g(y + [ν]) ≤ ∫ g(y + z)ν(dz) ≤ g(y) + ⟨ν, g^{∞}⟩ = g(y) .

If [ν] = 0, we deduce: g(y + [ν]) = ⟨ν, g(y + .)⟩, hence by the strict convexity of g (see (ii)) ν would be the Dirac mass at 0.

Acknowledgements. We would like to thank the referees for their very thorough reading of the paper, and for their many comments and suggestions, from which the final version of the paper greatly benefited.

References

[1] J.M. Ball: A version of the fundamental theorem for Young measures, Lecture Notes in Physics (PDE’s and continuum models of phase transitions, vol. 344, Springer (1989) 207– 215.

[2] M. Valadier: Young Measures, a) Methods of Nonconvex analysis (A. Cellina Ed.), Lecture notes in Math. vol. 1446, Springer (1990) 152–188.

b) A course on Young measures, Workshop on Measure Theory and Real Analysis, Grado, Prépublication 8-1994, Montpellier, 1993.

[3] J.M. Ball, F. Murat: Remarks on Chacon’s biting lemma, Proc. Amer. Math. Soc. 107 (1989) 655–663.

[4] J.M. Ball, K.W. Zhang: Lower semicontinuity of multiples integrals and the biting lemma, Proc. Royal Soc. Edinburgh 114A (1990) 367–379.

[5] D. Kinderlehrer, P. Pedregal: Remarks about gradient Young measures generated by sequences in Sobolev spaces, Preprint CMU, 1993.

[6] R.J. Di Perna, A.J. Majda: Oscillations and concentrations in weak solutions of the incompressible fluid equations, Com. Math. Phys. 108 (1987) 667–689.

[7] R.J. Di Perna, A.J. Majda: Reduced Hausdorff dimension and concentration cancellation for two-dimensional incompressible flow, J. Amer. Math. Soc. 1 (1988) 59–95.

[8] C. Goffman, J. Serrin: Sublinear functions on measures and variational integrals, Duke Math. J. 31 (1964) 159–178.

[9] Yu.G. Reshetnyak: Weak convergence of completely additive vector functions on a set. Siberian Math. J. 9 (1968) 1039–1045. (translation of Sibirsk. Mat. Z. 9 (1968) 1386–1394).

[10] G. Bouchitté, M. Valadier: Integral Representation of Convex Functionals on a Space of Measures, J. Functional Analysis, Vol. 80,2 (1988) 398–420.

[11] H. Brezis: Convergence in D′ and in L^{1} under strict convexity, to appear.

[12] S. Müller: Higher integrability of determinants and weak convergence in L^{1}, J. Reine Angew. Math. 412 (1990) 20–34.

[13] R.M. Dudley: Real analysis and probability, Wadsworth&Brooks/Cole Mathematics Series, California, 1989.

[14] H. Berliocchi, J.M. Lasry: Intégrandes normales et mesures paramétrées en calcul des variations, Bull. Soc. Math. France 101 (1973) 129–184.

148 J.J.Alibert, G. Bouchitté / Non-uniform integrability and generalized young measures HIER :

Leere Seite