New York Journal of Mathematics

New York J. Math. 3 (1997) 48–53.

A Refinement of Ball’s Theorem on Young Measures

Norbert Hungerbühler

Abstract. For a sequence u_{j} : Ω ⊂ ℝ^{n} → ℝ^{m} generating the Young measure ν_{x}, x ∈ Ω, Ball’s Theorem asserts that a tightness condition preventing mass in the target from escaping to infinity implies that ν_{x} is a probability measure and that f(u_{k}) ⇀ ⟨ν_{x}, f⟩ in L^{1} provided the sequence is equiintegrable. Here we show that Ball’s tightness condition is necessary for the conclusions to hold and that in fact all three, the tightness condition, the assertion ∥ν_{x}∥ = 1, and the convergence conclusion, are equivalent. We give some simple applications of this observation.

Contents

1. Introduction 48 2. Proof of Theorem 2 49 3. Applications 50

References  53

1. Introduction

Young measures have in recent years become an increasingly indispensable tool in the calculus of variations and in the theory of nonlinear partial differential equations (see, e.g., [5] or [2]. For a list of references for general Young measure theory see, e.g., [6]). In [1], Ball stated the following version of the fundamental theorem of Young measures, tailored for applications in these fields:

Theorem 1. Let Ω ⊂ ℝ^{n} be Lebesgue measurable, let K ⊂ ℝ^{m} be closed, and let u_{j}: Ω → ℝ^{m}, j ∈ ℕ, be a sequence of Lebesgue measurable functions satisfying u_{j} → K in measure as j → ∞, i.e., given any open neighbourhood U of K in ℝ^{m}

lim_{j→∞} |{x ∈ Ω: u_{j}(x) ∉ U}| = 0.

Then there exists a subsequence u_{k} of u_{j} and a family (ν_{x}), x ∈ Ω, of positive measures on ℝ^{m}, depending measurably on x, such that

(i) ∥ν_{x}∥_{M} := ∫_{ℝ^{m}} dν_{x} ≤ 1 for a.e. x ∈ Ω,

\begin{matrix}c \\ ◯ \end{matrix}1997 State University of New York

ISSN 1076-9803/97

Received April 9, 1997.

Mathematics Subject Classification. 46E27, 28A33, 28A20.

Key words and phrases. Young measures.

(ii) spt ν_{x} ⊂ K for a.e. x ∈ Ω, and (iii) f(u_{k}) ⇀^{∗} ⟨ν_{x}, f⟩ = ∫_{ℝ^{m}} f(λ)dν_{x}(λ) in L^{∞}(Ω) for each continuous function

f: ℝ^{m} → ℝ satisfying lim_{|λ|→∞} f(λ) = 0. Suppose further that {u_{k}} satisfies the boundedness condition

∀R > 0: lim_{L→∞} sup_{k∈ℕ} |{x ∈ Ω ∩ B_{R}: |u_{k}(x)| ≥ L}| = 0, (1)

where B_{R} = B_{R}(0). Then

∥ν_{x}∥_{M} = 1 for a.e. x ∈ Ω (2)

(i.e., ν_{x} is a probability measure), and the following condition holds:

{For any measurable A ⊂ Ω and any continuous function f: ℝ^{m} → ℝ such that {f(u_{k})} is sequentially weakly relatively compact in L^{1}(A) we have f(u_{k}) ⇀ ⟨ν_{x}, f⟩ in L^{1}(A).

(3)

Improved versions of this theorem can be found, e.g., in [4]. The aim of this note is to prove that (1) is necessary for (2) and (3) to hold, and that in fact (1), (2) and (3) are equivalent. We will give some simple consequences of this fact.

Theorem 2. Let Ω, u_{j} and ν_{x} be as in Theorem 1. Then (1), (2) and (3) are equivalent.

Remarks: (a) It was proved in [1] that (1) is equivalent to the following condition: Given any R > 0 there exists a continuous nondecreasing function g_{R}: [0, ∞) → ℝ, with lim_{t→∞} g_{R}(t) = ∞, such that

sup_{k∈ℕ} ∫_{Ω∩B_{R}} g_{R}(|u_{k}(x)|)dx < ∞.

(b) In [1] it is also shown, that under hypothesis (1) for any measurable A ⊂ Ω

f(·, u_{k}) ⇀ ⟨ν_{x}, f(x, ·)⟩ in L^{1}(A)

for every Carathéodory function f: A×ℝ^{m} → ℝ such that {f(·, u_{k})} is sequentially weakly relative compact in L^{1}(A). Hence, this fact is also equivalent to (1), (2) and (3).

(c) Ball also shows in [1], that if u_{k} generates the Young measure ν_{x}, then for ψ ∈ L^{1}(Ω; C_{0}(ℝ^{m}))

lim_{k→∞} ∫_{Ω} ψ(x, u_{k}(x))dx = ∫_{Ω}⟨ν_{x}, ψ(x, ·)⟩dx.

Here, C_{0}(ℝ^{m}) denotes the Banach space of continuous functions f: ℝ^{m} → ℝ satisfying lim_{|λ|→∞} f(λ) → 0 equipped with the L^{∞}-norm.

2. Proof of Theorem 2

First we prove (2) ⟹ (1). We assume by contradiction that (2) holds and that there exists R > 0 and ε > 0 with the following property: There exists a sequence L_{i} → ∞ and integers k_{i} such that |{x ∈ Ω ∩ B_{R}: |u_{k_{i}}(x)| ≥ L_{i}}| > ε for all i ∈ ℕ. For ρ > 0 consider the function

α_{ρ}(t) := \begin{cases}1 & if t ≤ ρ \\ 0 & if t ≥ ρ + 1 \\ ρ + 1 − t & if ρ < t < ρ + 1 \end{cases}

Then φ_{ρ}: ℝ^{m} → ℝ, x ↦ α_{ρ}(|x|), is in C_{0}(ℝ^{m}). Hence, applying the first part of Theorem 1, we have that

lim_{k→∞} ∫_{Ω} φ_{ρ}(u_{k})χ_{B_{R}}dx = ∫_{Ω} ∫_{ℝ^{m}} φ_{ρ}(λ)dν_{x}(λ)χ_{B_{R}}dx. (4)

Notice that k_{i} → ∞ for i → ∞ since the functions u_{j} are finite for a.e. x ∈ Ω. Hence, u_{k_{i}} is a subsequence of u_{k} and for i large enough, we find

|Ω ∩ B_{R}| − ε ≥ ∫_{Ω} φ_{ρ}(u_{k_{i}})χ_{B_{R}}dx.

Thus, (4) implies

|Ω ∩ B_{R}| − ε ≥ ∫_{Ω} ∫_{ℝ^{m}} φ_{ρ}(λ)dν_{x}(λ)χ_{B_{R}}dx. (5)

On the other hand, by the monotone convergence theorem, we conclude that the right hand side of (5) converges for ρ → ∞ to

∫_{Ω} ∫_{ℝ^{m}} dν_{x}(λ)χ_{B_{R}}dx = ∫_{Ω} ∥ν_{x}∥_{M}χ_{B_{R}}dx = |Ω ∩ B_{R}|

by (2) and this contradicts (5).

Second we prove that (3) ⟹ (2). Let R > 0 be fixed and let f denote the function constant 1 on ℝ^{m}. Then f(u_{j}) is sequentially weakly relative compact on Ω ∩ B_{R} and (3) implies

(6) |Ω ∩ B_{R}| = ∫_{Ω∩B_{R}} f(u_{k})χ_{B_{R}}dx →

→ ∫_{Ω∩B_{R}} ∫_{ℝ^{m}} f(λ)dν_{x}(λ)χ_{B_{R}}dx = ∫_{Ω∩B_{R}} ∥ν_{x}∥_{M}dx.

Since ∥ν_{x}∥_{M} ≤ 1 by (i) in Theorem 1, we conclude that ∥ν_{x}∥_{M} = 1 for a.e. x ∈ Ω ∩ B_{R}. Since R was arbitrary, the claim follows.

3. Applications

The following propositions are certainly known to the experts in the field, but we want to show that the sharp version of Ball’s theorem which we now have at our disposal, considerably simplifies the proofs.

Proposition 1. If |Ω| < ∞ and ν_{x} is the Young measure generated by the (whole) sequence u_{j} then

u_{j} → u in measure ⟺ ν_{x} = δ_{u(x)} for a. e. x ∈ Ω.

A precursor of this proposition can be found, e.g., in [3] (see also [4]).

Proof. Let us first assume that u_{j} → u in measure, i.e., ∀ε > 0 we have

lim_{j→∞} |{|u_{j} − u| > ε}| = 0. (7)

For φ ∈ C_{c}^{∞}(ℝ^{m}) and ζ ∈ L^{1}(Ω),

|∫_{Ω} ζ (φ(u_{j}) − φ(u))dx| ≤

|∫_{|u_{j}−u|>ε} ζ (φ(u_{j}) − φ(u))dx| + |∫_{|u_{j}−u|≤ε} ζ (φ(u_{j}) − φ(u))dx| =: I + II.

By choosing ε appropriately, we can make II as small as we want, since we observe that II ≤ ε∥Dφ∥_{L^{∞}}∥ζ∥_{L^{1}}. For I we then have

I ≤ 2∥φ∥_{L^{∞}} ∫_{|u_{j}−u|>ε} |ζ|dx

which converges to 0 as j tends to ∞ by absolute continuity of the integral and (7). Since C_{c}^{∞} is dense in C_{0} we conclude that for all φ ∈ C_{0}

φ(u_{j}) ⇀^{∗} ⟨δ_{u(x)}, φ⟩ in L^{∞}(Ω)

and hence ν_{x} = δ_{u(x)}.

Now we assume ν_{x} = δ_{u(x)}, hence (2) is fulfilled. First step: We consider the case that u_{j} is bounded in L^{∞}. Then by (3) we conclude that for φ(x) := |x|^{2}

∥u_{j}∥_{L^{2}}^{2} = ∫_{Ω} φ(u_{j})dx → ∫_{Ω} φ(u)dx = ∥u∥_{L^{2}}^{2} (8)

for j → ∞. On the other hand choosing φ = id we similarly find that u_{j} ⇀ u weakly in L^{2}(Ω), which in combination with (8) gives that u_{j} → u in L^{1}(Ω). Thus for all α > 0 we have

α|{|u_{j} − u| ≥ α}| ≤ ∫_{Ω} |u_{j} − u|dx → 0

as j → ∞, and hence u_{j} → u in measure.

Second step: We show that if u_{j} generates the Young measure δ_{u(x)} then T_{R}(u_{j}) → T_{R}(u) in measure, where T_{R} denotes the truncation T_{R}(x) := x min\{1, \frac{R}{|x|}\}, R > 0 fixed. In fact, for f ∈ C_{0}(ℝ^{m}) we have that f ∘ T_{R} is continuous and f(T_{R}(u_{j})) is equiintegrable (and hence, by the Dunford-Pettis theorem, sequentially weakly precompact in L^{1}(Ω)). Since (2) is fulfilled, we conclude by (3) that for ζ ∈ L^{∞}(Ω)

∫_{Ω} ζ f(T_{R}(u_{j}))dx → ∫_{Ω} ζ f(T_{R}(u))dx.

This implies that T_{R}(u_{j}) generates the Young measure δ_{T_{R}(u(x))} and by the first step, the claim follows.

Third step: We show, that u_{j} → u in measure. Let ε > 0 be given. Then we have:

|{|u_{j} − u| > ε}| ≤ |{|u_{j} − u| > ε, |u| ≤ R, |u_{j}| ≤ R}| + |{|u| > R}| + |{|u_{j}| > R}|
=: I + II + III.

II can be made arbitrarily small by choosing R > 0 large enough. By (2) we have (1) which implies that III is, again for R large enough, uniformly in j as small as we want. Finally by the second step, I → 0 for j → ∞. □

Our second application is the following proposition:

Proposition 2. Let |Ω| < ∞. If the sequences u_{j}: Ω → ℝ^{m} and v_{j}: Ω → ℝ^{k} generate the Young measures δ_{u(x)} and ν_{x} respectively, then (u_{j}, v_{j}) generates the Young measure δ_{u(x)} ⊗ ν_{x}.

This result also holds for sequences μ_{j}, λ_{j} of Young measures converging in the narrow topology to μ and λ respectively: see [6]. However it is false if both μ and λ are not Dirac measures. E.g., consider the Rademacher functions u_{1}(x) := (−1)^{⌊x⌋} and u_{n}(x) = u_{1}(nx). u_{n} and −u_{n} generate the Young measure \frac{1}{2}(δ_{−1} + δ_{1}), but (u_{n}, u_{n}) and (−u_{n}, u_{n}) obviously generate different measures (consider the sets K = {(−1, −1), (1, 1)} and K = {(−1, 1), (1, −1)} respectively in Theorem 1).

Proof of Proposition 2. We have to show that for all φ ∈ C_{c}^{∞}(ℝ^{m} × ℝ^{k}) there holds φ(u_{j}, v_{j}) ⇀^{∗} ∫_{ℝ^{k}} φ(u(x), λ)dν_{x}(λ). So, let ζ ∈ L^{1}(Ω). We have

|∫_{Ω} ζ (φ(u_{j}, v_{j}) − ∫_{ℝ^{k}} φ(u, λ)dν_{x}(λ))dx| ≤

≤ |∫_{|u_{j}−u|<ε} ζ (φ(u_{j}, v_{j}) − φ(u, v_{j}))dx| + |∫_{|u_{j}−u|≥ε} ζ (φ(u_{j}, v_{j}) − φ(u, v_{j}))dx|

+ |∫_{Ω} ζ (φ(u, v_{j}) − ∫_{ℝ^{k}} φ(u, λ)dν_{x}(λ))dx| =: I + II + III.

Since I ≤ ε∥ζ∥_{L^{1}(Ω)}∥Dφ∥_{L^{∞}}, the first term is small for ε > 0 small. For ε > 0 fixed, we have for j → ∞

II ≤ 2∥φ∥_{L^{∞}} ∫_{|u_{j}−u|≥ε} |ζ|dx → 0

since by Proposition 1 the sequence u_{j} converges to u in measure. Since L^{∞}(Ω) is dense in L^{1}(Ω) we may assume that ζ ∈ L^{∞}(Ω). Thus, the function ζ(x)φ(u(x), ·) is in L^{1}(Ω, C_{0}(ℝ^{k})) and hence III → 0 as j → ∞ by Remark (c). □

The third application we consider is a criterion for the pointwise convergence of Fourier series, which is similar to Dini’s test (see [7]).

Theorem 3. Let f ∈ L_{loc}^{1}(ℝ) be a 2π periodic complex function. If z ∈ ℝ is a point with the property that

∫_{−π}^{π} |\frac{f(x) − f(z)}{x − z}| dx < ∞ (9)

then the Fourier series of f converges in z to f(z).

Proof. With

D_{N}(x) := \frac{sin(N + \frac{1}{2})x}{sin \frac{x}{2}}

the Nth Fourier approximation of f is s_{N}(z) = \frac{1}{2π} ∫_{−π}^{π} f(z−x)D_{N}(x)dx. The Young measure generated by the (whole) sequence sin(N + \frac{1}{2})x is a probability measure ν_{x} with vanishing first moment ⟨ν_{x}, id⟩ = 0 (see, e.g., [6]). Hence, the Young measure μ_{x} generated by the sequence (f(z − x) − f(z))D_{N}(x) also has these properties for a. e. x. Now, (9) implies that the sequence (f(z −x)−f(z))D_{N}(x) is equiintegrable

On Ball’s Theorem on Young measures 53

and hence (since μ_{x} is a probability measure for a. e. x) by equivalence of (2) and (3), we have as N → ∞

s_{N}(z) = \frac{1}{2π} ∫_{−π}^{π}(f(z − x) − f(z))D_{N}(x)dx + \frac{f(z)}{2π} ∫_{−π}^{π} D_{N}(x)dx → f(z)

since the first term converges to zero and the second term equals f(z) for all N. □

References

[1] J. M. Ball, A version of the fundamental theorem for Young measures, Partial Differential Equations and Continuum Models of Phase Transitions (Proceedings of an NSF-CNRS joint seminar), Springer-verlag, Berlin, 1989.

[2] G. Dolzmann, N. Hungerbühler, S. Müller, Non-linear elliptic systems with measurevalued right hand side, Math. Z., to appear.

[3] L. C. Evans, Weak Convergence Methods for Nonlinear Partial Differential Equations, CBMS Regional Conf. Ser. in Math., no. 74, Amer. Math. Soc., Providence, RI, 1990.

[4] J. Kristensen, Lower Semicontinuity of Variational Integrals, Ph.D. Thesis, Technical University of Denmark, 1994.

[5] L. Tartar, The compensated compactness method applied to systems of conservation laws, Systems of Nonlinear Partial Differential Equations (Oxford, 1982, J. M. Ball, ed.), NATO Adv. Sci. Inst. Ser. C: Math. Phys. Sci., no. 111, Reidel, Dordrecht-Boston, 1983, pp. 263– 285.

[6] M. Valadier, A course on Young measures, Workshop on Measure Theory and Real Analysis (Grado, 1993), Rend. Istit. Mat. Univ. Trieste 26 (1994), suppl., 349–394.

[7] A. Zygmund, Trigonometric Series, Cambridge University Press, Cambridge, 1977.

Departement Mathematik, ETH-Zentrum, CH-8092 Zürich (Switzerland) buhler@math.ethz.ch, http://www.math.ethz.ch/˜ buhler/noebi.html