C O L L O Q U I U M M A T H E M A T I C U M VOL. 74 1997 NO. 1

C O L L O Q U I U M M A T H E M A T I C U M

VOL. 74 1997 NO. 1

ON LOWER SEMICONTINUITY OF MULTIPLE INTEGRALS

BY

AGNIESZKA K A L A M A J S K A (WARSZAWA)

We give a new short proof of the Morrey–Acerbi–Fusco–Marcellini Theo- rem on lower semicontinuity of the variational functional

T

ΩF (x, u, ∇u) dx.

The proofs are based on arguments from the theory of Young measures.

1. Introduction and statement of results. Let Ω be a bounded open domain in Rⁿ. Define the functional

(1) I(u) =

\

Ω

F (x, u, ∇u) dx for u ∈ W^1,p(Ω, R^m).

Such functionals are related to questions of nonlinear elasticity and Skyrme’s model for meson fields and have been investigated by many authors (see e.g. [1], [2], [4], [6], [10]–[17], [19], [20], [22], [23]).

We give a short proof of the following theorem due to Morrey, Acerbi, Fusco, and Marcellini (see [22], [1], [19]; the definition of quasiconvexity is given in Section 2).

Theorem 1.1. Let Ω ⊆ Rⁿ be a bounded domain, 1 ≤ p ≤ ∞, and let F : Ω × R^m× R^mn → [0, ∞] satisfy

(i) F (x, s, λ) is a Carath´eodory function (i.e. measurable in x ∈ Ω and continuous in(s, λ) ∈ R^m× R^mn),

(ii) there exists a Carath´eodory function E(·, ·) such that, for almost every x and all (s, λ), |F (x, s, λ)| ≤ E(x, s)g(λ) if p = ∞, for some contin- uous function g, and |F (x, s, λ)| ≤ E(x, s)(1 + |λ|^p) if p < ∞,

(iii) for almost every x and all s, the mapping λ 7→ F (x, s, λ) is quasi- convex.

If u^j → u in L^p(Ω, R^m) and ∇u^j ⇀ ∇u in L^p(Ω, R^mn) as j → ∞ (∇u^j ⇀ ∇u in L^{∗ ∞}(Ω, R^mn) if p = ∞) then the functional (1) satisfies

(2) I(u) ≤ lim inf

j→∞ I(u^j).

1991 Mathematics Subject Classification: Primary 49J45.

[71]

Our proof of Theorem 1.1 is based on the theory of Young measures, and will be obtained as an easy consequence of the following Jensen-type inequalities for Young measures.

Theorem 1.2. Let Ω ⊆ Rⁿ be any bounded domain and 1 ≤ p ≤ ∞.

Suppose that {ν_x}x∈Ω is the Young measure (see Definition 3.2) generated by the sequence ∇u^j where u^j ∈ W^1,p(Ω, R^m), and ∇u^j ⇀ ∇u in L^p(Ω, R^mn) asj → ∞ (∇u^j ⇀ ∇u in L^{∗ ∞}(Ω, R^mn) if p = ∞). If F : Ω ×R^mn → [−∞, ∞]

satisfies

(i) F (x, λ) is a Carath´eodory function (for x ∈ Ω, λ ∈ R^m_n),

(ii) there exists a mesurable function E(·) such that, for almost every x and all λ, |F (x, λ)| ≤ E(x)g(λ) if p = ∞, for some continuous function g, and |F (x, λ)| ≤ E(x)(1 + |λ|^p) if p < ∞,

(iii) the mapping λ 7→ F (x, λ) is quasiconvex for almost every x,

then the following Jensen-type inequality is satisfied for almost everyx ∈ Ω:

(3) F

x,

\

R^m_n

λ νx(dλ)

≤

\

R^m_n

F (x, λ) νx(dλ)

and ∇u(x) =

T

R^m_n λ ν_x(dλ).

It is known that Theorems 1.1 and 1.2 are equivalent (see e.g. [6], [15]–

[17]), but as far as I know a direct proof of Theorem 1.2 is missing in the literature. The known proof of Theorem 1.2 requires Theorem 1.1, or its slightly less general version due to Acerbi and Fusco [1]. Theorem 1.1 in the formulation given here was obtained by Marcellini [19]. He did not use Young measures, but the proof was rather long. We want to show that a direct application of Young measures is a useful tool and can abbreviate the already known reasonings.

2. Preliminaries and notation. We use standard notation for the well known function spaces W^1,p(Ω) (Sobolev space), C0(R^l) (continuous func- tions vanishing at infinity), C(Ω) (continuous functions), Lip(Ω) (Lipschitz functions), and M(Ω) (Radon measures). If f ∈ C(Ω) and µ ∈ M(Ω), then (f, µ) will stand for

T

Ωf (λ) µ(dλ). We write

4

Af dx for |A|⁻¹

T

Af dx. We denote by Q(x, r) the cube with center x and edges of length r. If x_n, x are elements of a Banach space then we denote by xn → x the strong (norm) convergence, by xn ⇀ x the weak convergence and by xn

⇀ x the weak ∗∗

convergence. By C we denote the general constant, which can change even in the same line.

The following theorem is well known and has many extensions (see e.g.

[9, Theorem 13], [18], [21], [7], [8]).

Theorem 2.1. Let Ω ⊆ Rⁿ be an open set and 1 ≤ p < ∞. Then for any u ∈ W^1,p(Ω) and any λ > 0 there exists a closed set Fλ ⊆ Ω and a mapping u_λ ∈ Lip(Ω) such that

(i) λ^p|Ω \ Fλ| → 0 as λ → ∞,

(ii) ∇u = ∇uλ for almost every x ∈ Fλ,

(iii) |∇u_λ(x)| < Cλ for almost every x ∈ Ω, with C independent of x and λ,

(iv) k∇u − ∇uλk_L^p(Ω)→ 0 as λ → ∞.

We recall the fundamental theorem of Young (see [3]).

Theorem 2.2. Let Ω ⊆ Rⁿ be a measurable bounded set. Assume that u^j : Ω → R^m, j = 1, 2, . . . , is a sequence of measurable functions satisfying the following tightness condition:

sup

j

|{x ∈ Ω : |u^j(x)| ≥ k}| → 0 as k → ∞.

Then there exists a subsequence {u^k} and a family {νx}x∈Ω of probability measures, ν_x ∈ M(R^m), such that

(i) for every f ∈ C0(R^m) the function x 7→ (f, νx) is measurable, (ii) if K ⊆ Rⁿ is a closed set, and for every j and almost every x, u^j(x) ∈ K, then supp ν_x⊆ K for almost every x,

(iii) if A ⊆ Ω is measurable and f : Ω × R^m→ R satisfies

• f is a Carath´eodory function,

• the sequence {f (x, u^k(x))} is sequentially weakly relatively com- pact in L¹(A),

then {f (x, u^k(x))} converges weakly in L¹(A) to f given by f (x) =

\

R^m

f (x, λ) ν_x(dλ).

Definition 2.1. We say that u^j generates the Young measure {νx}x∈Ω

if {νx}x∈Ω satisfies (i) and for any f ∈ C0(R^m), f (u^j) ⇀ f = (f, ν^∗ x) in L^∞(Ω).

The following useful fact is a generalization of that given in [6, Lemma 2.2]. Although this form is not required for our needs, for completeness, and to show some particular techniques, we try to give a possibly general formulation and include a detailed proof.

Theorem 2.3. Suppose that u^j = (w^j, v^j) : Ω → R^m× R^k satisfy the tightness condition and generate the Young measure{µ_x}x∈Ω. Suppose fur- ther that w^j → w in measure and that {v^j}j∈N generates the Young mea- sure{ν_x}_x∈Ω. Then for almost everyx ∈ Ω we have µ_x= δ_w(x)⊗ ν_x, which means that for any f ∈ C0(R^m× R^k) and almost every x ∈ Ω,

(4)

\

R^m×R^k

f (s, λ) µ_x(ds, dλ) =

\

R^k

f (w(x), λ) ν_x(dλ).

If f is a Carath´eodory function on Ω × (R^m × R^k) and |f (z, s, λ)| ≤ C(z)g(s, λ) with some measurable function C and a continuous function g such that for almost all x ∈ Ω,

(5)

\

R^m×R^k

g(s, λ) µ_x(ds, dλ) < ∞ and

\

R^k

g(w(x), λ) ν_x(dλ) < ∞, then for almost every x ∈ Ω,

(6)

\

R^m×R^k

f (x, s, λ) µx(ds, dλ) =

\

R^k

f (x, w(x), λ) νx(dλ).

P r o o f. Let f ∈ C0(R^m× R^k) and set h^j = f (w^j, v^j) − f (w, v^j). Since f is uniformly continuous, it follows that h^j → 0 in measure, and moreover

|h^j| ≤ 2kf k_L^∞_(R^m_×R^k₎. Thus, by the Lebesgue Dominated Convergence Theorem we have h^j → 0 strongly in L¹(Ω), while on the other hand, by Theorem 2.2 it converges to (f, µ_x)−(f (w(x), ·), ν_x) weakly in L¹(Ω). Hence (f, µx) − (f (w(x), ·), νx) = 0 almost everywhere, from which (4) follows. To prove (6) we consider three cases: 1) f does not depend on x, 2) C(z) ≤ K < ∞ and f is continuous on Ω × R^m× R^k, and 3) the general case.

In the first case define φ : [0, ∞) → R by φ(t) = 1 on [0, 1], φ(t) = −t + 2 on [1, 2] and φ(t) = 0 for t > 2. Since f^j(s, λ) = f (s, λ)φ(|(s, λ)|/j) ∈ C0(R^m× R^k), it follows that the formula

(7)

\

R^m×R^k

f^j(s, λ) µx(ds, dλ) =

\

R^k

f^j(w(x), λ) νx(dλ)

holds everywhere on a set Ω(j) of full measure. In particular, (7) holds everywhere on the set eΩ = T

jΩ(j), which is still of full measure. We can assume additionally that (5) holds for all x ∈ eΩ. Since |f^j| ≤ |f |, we can let j tend to infinity, apply the Lebesgue Dominated Convergence Theorem, and verify that (6) holds everywhere on eΩ.

In the second case choose a dense countable subset {B^j} ⊆ Ω and con- sider the functions F^j(s, λ) = f (B^j, s, λ). Since by Case 1 the equality (6) is satisfied with f = F^j on a set Ω1(j) of full measure we see that

\

R^m×R^k

f (B^j, s, λ) µx(ds, dλ) =

\

R^k

f (B^j, w(x), λ) νx(dλ) on the set Ω1=T

jΩ1(j), which is still of full measure and does not depend on j. Take an arbitrary x ∈ Ω₁ and a sequence B^jk → x as k → ∞. Now it suffices to check that by the Lebesgue Dominated Convergence Theorem the left hand side of the equality tends to

T

R^m×R^kf (x, s, λ) µx(ds, dλ), while the right hand side tends to

T

R^kf (x, w(x), λ) νx(dλ).

Finally, in the last case we use the Scorza Dragoni Theorem and Lusin Theorem (see e.g. [13]) and bite off sets Ωεof arbitrarily small measure such that f is continuous on (Ω \ Ω_ε) × R^m× R^k and C is bounded on Ω \ Ω_ε. Thus (6) is satisfied almost everywhere on Ω \ Ω_ε, and hence it is satisfied almost everywhere on Ω.

Let us state Chacon’s Biting Lemma (see e.g. [5]).

Theorem 2.4 [Biting Lemma]. Let Ω ⊆ Rⁿ, |Ω| < ∞ and suppose that {f^j} is a bounded sequence in L¹(Ω). Then there exists a subsequence {f^ν}, a function f ∈ L¹(Ω) and a decreasing family of measurable sets Ek such that |E_k| → 0 as k → ∞ and for any k,

f^ν⇀ f in L¹(Ω \ E_k) as ν → ∞.

Definition 2.2 (see e.g. [24]). We will say that {f^j} converges to f in the biting sense (f^j ⇀ f ) whenever there is a set E of arbitrarily small^b measure such that f^j ⇀ f in L¹(Ω \ E).

Finally, we recall the known definition of quasiconvexity.

Definition 2.3. The function F : R^mn → R is quasiconvex if for any A ∈ R^mn, any cube Q ⊆ Rⁿ and any φ ∈ C₀^∞(Q, R^m),

(8) F (A) ≤

<

Q

F (A + ∇φ) dx.

3. Proofs of Theorems 1.1 and 1.2

P r o o f o f T h e o r e m 1.2. We can assume that Ω is a ball (if h1≤ h2

almost everywhere on every ball B ⊆ Ω then h₁≤ h₂almost everywhere on Ω). If we take f (λ) = λ, f : R^mn → R^mn and apply the Young Theorem to every coordinate of f (∇u^j) we immediately derive

T

R^m_n λ νx(dλ) = ∇u(x), for almost every x.

We distinguish the following cases: 1) F = F (λ) and p = ∞, 2) F = F (λ) and 1 ≤ p < ∞, and 3) the general case.

C a s e 1. Let x ∈ Ω and r > 0 be such that Q(x, r) ⊆ Ω. Take 0 < σ < r and choose φ_σ ∈ C₀^∞(Q(x, r)), φ_σ ≡ 1 on Q(x, r − σ). By standard arguments the function w^j_σ= φσ(u^j− u) can be substituted in (8).

That gives for arbitrary A ∈ R^mn, F (A) ≤

<

Q(x,r)

F (A + ∇φσ· (u^j− u) + φσ(∇u^j − ∇u)) dy = I(x, r, σ, j).

Since {F (A+∇w^j_σ)}_j is relatively compact in L¹(Ω), by the Young Theorem applied to the sequence (u^j − u, ∇u^j) and by Theorem 2.3,

I(x, r, σ, j) →

<

Q(x,r)

\

R^m

n

F (A + φ_σ(y)(λ − ∇u(y))) ν_y(dλ) dy = I(x, r, σ) as j → ∞ and ν_y is supported on a bounded set. Hence, if we apply the Lebesgue Dominated Convergence Theorem and let σ → 0, we see that

I(x, r, σ) →

<

Q(x,r)

\

R^m_n

F (A + λ − ∇u(y)) νy(dλ) dy.

By the Lebesgue Differentiation Theorem for any A ∈ R^mn there is a set Ω(A) ⊆ Ω such that |Ω \ Ω(A)| = 0, and for each x ∈ Ω(A),

(9) F (A) ≤

\

R^m_n

F (A + λ − ∇u(x)) ν_x(dλ).

We can additionally assume that |∇u(x)| < ∞ for every x ∈ Ω(A). Let {A^j} be a countable dense subset in R^mn. Since Ω1=T

jΩ(A^j) is still of full measure in Ω, for every x ∈ Ω1 the inequality (9) is satisfied with A = A^j, for arbitrary j. Take x ∈ Ω1 and let A^jk → ∇u(x) as k → ∞. Now it suffices to note that F (A^jk) → F (∇u(x)), and

\

R^m_n

F (A^jk − ∇u(x) + λ) ν_x(dλ) →

\

R^m_n

F (λ) ν_x(dλ).

C a s e 2. We apply Theorem 2.1 and find u_λ such that k∇u_λk_L^∞_(Ω) ≤ Cλ and ∇uλ = ∇u almost everywhere on Fλ, where λ^p|Ω \ Fλ| → 0 as λ → ∞. Let λ_k = k, and let {ν_x^k} be the Young measure generated by a subsequence of {∇u^j_k}_j. Note that for any k we have F (λ, ν_x^k) ≤ (F, ν_x^k) almost everywhere. Thus it suffices to apply the following lemma.

Lemma 3.1. Let f = f (λ) with |f (λ)| ≤ C(1 + |λ|^p), and {νx}x∈Ω and {ν_x^k}x∈Ω be as above. Then for everyε > 0 we can find a set E ⊆ Ω such that |E| < ε and (f, ν_x^k) → (f, νx) in L¹(Ω \ E) as k → ∞.

P r o o f. Take ε > 0. According to Theorems 2.4 and 2.2 we find a set E ⊆ Ω such that |E| < ε and f (∇u^j) ⇀ (f, ν_x) weakly in L¹(Ω \ E). We have

\

Ω\E

|(f, ν_x^k) − (f, νx)| dx

= sup

kφkL∞(Ω)≤1

\

Ω\E

φ(x)((f, ν_x^k) − (f, νx)) dx 

= sup

kφkL∞(Ω)≤1

  lim

j→∞

\

Ω\E

φ(x)(f (∇u^j_k) − f (∇u^j)) dx 

≤ sup

j

\

(Ω\E)\Fk

|f (∇u^j_k)| + sup

j

\

(Ω\E)\Fk

|f (∇u^j)| dx^k→∞−→ 0.

The convergence follows from the estimates on f and the Dunford–Pettis Theorem.

C a s e 3. We use exactly the same arguments as in the proof of Theorem 2.3, cases 2 and 3.

P r o o f o f T h e o r e m 1.1. Suppose that {u^j}j∈Nsatisfies the assump- tions of Theorem 1.1. Let α = lim inf_j→∞I(u^j). If α = ∞ the assertion is satisfied. Suppose that α < ∞. In this case the sequence {F (x, u^j, ∇u^j)}j∈N

is bounded in L¹(Ω). By Theorems 2.2–2.4 we find a subsequence {u^l} with the properties: 1) I(u^l) → α as l → ∞, 2) the sequence {∇u^l} generates the Young measure {ν_x}x∈Ω, 3) there exists a family {E_k} of sets such that |Ek| → 0 and {F (x, u^l, ∇u^l)}l is weakly convergent in L¹(Ω \ Ek) to

T

R^m_n F (x, u(x), λ) ν_x(dλ).

Since F_u(x, λ) = F (x, u(x), λ) satisfies the assumptions of Theorem 1.2, we have

T

R^m_n F (x, u(x), λ) νx(dλ) ≥ F (x, u(x), ∇u(x)) for almost every x.

Now it suffices to note that α = lim

l→∞

\

Ω

F (x, u^l, ∇u^l) dx ≥ lim

l→∞

\

Ω\Ek

F (x, u^l, ∇u^l) dx

=

\

Ω\Ek

\

R^m_n

F (x, u(x), λ) νx(dλ) ≥

\

Ω\Ek

F (x, u(x), ∇u(x)) dx.

R e m a r k 3.1. It has been proved by Kristensen [17] that the Jensen inequalities of Theorem 1.2 can be generalized to a certain class of functions which are Borel measurable with respect to the last variable.

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Institute of Mathematics University of Warsaw Banacha 2

00-097 Warszawa, Poland E-mail: kalamajs@mimuw.edu.pl

Received 1 April 1996;

revised 27 November 1996