Weak Convergence Methods for Nonlinear Partial Differential ...

2. u periodic with unit cell [0, 1] n , u (j) (x) = u(jx)
(so that u (j) ⇀ ∗ ∫
ū ≡ − u(y) dy), then
[0,1]
∫
f(u) ⇀ ∗ −
[0,1]
∫
f(u(x)) dx =
f(y) dν(y),
where ν (independent of x) is the image measure of the uniform distribution
under u.
Remark 2.69 1. One can show that any measurably parameterized measure
(ν x ) arises as the Young measure of a suitable sequence (u (j) ).
2. Uniqueness: Suppose, (ν x ), (˜ν x ) are two Young measures associated to
(u (j) ). Then for all continuous f:
∫
∫
f(y) dν x (y) dx = w ∗ -lim f(u (j) ) = f(y) d˜ν x dx for a.e. x
j
and thus
ν x = ˜ν x for a.e. x.
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