Weak Convergence Methods for Nonlinear Partial Differential ...

does the job:
∫ (∫
g(x, y)
f(x, y) ∫
E F g(x, y) dy dy g(x, y) dy dx
F
)∫F
∫
= f(x, y)g(x, y) dydx.
E×F
Proof of Theorem 2.65. To a measurable function v : Ω → K one associates a
non-negative measure ν ∈ M(Ω × R m ) via
∫
〈ν, ϕ(x, y)〉 M,C0
= ϕ(x, v(x)) dx ∀ ϕ ∈ C 0 (Ω × R m ).
(i.e., “ν = L n × δ v(x) ”). In particular, for v = u (j) we have
〈
ν (j) , ϕ 〉 ∫
M,C 0
= ϕ(x, u (j) (x)) dx.
This defines a bounded (by |Ω|) sequence of Radon measures on Ω × R m from
which we may extract a convergent subsequence (not relabeled)
The limiting measure ν satisfies
1. ν ≥ 0,
2. supp ν ⊂ Ω × K and
3. proj Ω ν = L n ,
because for ϕ ∈ C 0 (Ω × R m ):
ν (j) ∗ ⇀ ν in M(Ω × R m ).
1. ϕ ≥ 0 implies 〈ν, ϕ〉 = lim j
〈
ν (j) , u 〉 = lim j
∫
ϕ(x, u (j) (x)) dx ≥ 0.
2. ∀ ϕ such that ϕ ≡ 0 on Ω × K:
∫
〈ν, ϕ〉 = lim
j
ϕ(x, u (j) (x)) dx = 0.
3. ϕ(x, y) = ψ(x) implies
〈ν, ϕ〉 = lim
j
∫
∫
ψ(x) =
ψ(x) dx.
51