77.5. THE MAIN RESULT 2631
Hence, by Fatou’s lemma,
lim infε→0⟨Buε ,uε⟩(T ) = lim inf
ε→0
∞
∑i=1⟨B(uε (T )) ,ei⟩2
≥∞
∑i=1
lim infε→0⟨B(uε (T )) ,ei⟩2
=∞
∑i=1
lim infε→0⟨Buε (T ) ,ei⟩2
=∞
∑i=1⟨Bu(T ) ,ei⟩2
= ⟨B(u(T )) ,u(T )⟩= ⟨Bu,u⟩(T )
From 77.5.56, letting t = T,
lim supε→0⟨u∗ε ,uε⟩V ′,V ≤ lim sup
ε→0
(⟨ f ,uε⟩+
12⟨Bu0,u0⟩−
12⟨Buε ,uε⟩(T )
)≤ ⟨ f ,u⟩V ′,V +
12⟨Bu0,u0⟩−
12⟨Bu,u⟩(T ) = ⟨u∗,u⟩V ′,V
It follows thatlim sup
ε→0⟨u∗ε ,uε −u⟩ ≤ ⟨u∗,u⟩V ′,V −⟨u
∗,u⟩V ′,V = 0
and solim inf
ε→0⟨u∗ε ,uε − v⟩ ≥ ⟨u∗ (v) ,u− v⟩
for any v ∈ V where u∗ (v) ∈ A(u,ω). In particular for v = u. Hence
lim infε→0⟨u∗ε ,uε −u⟩ ≥ ⟨u∗ (v) ,u−u⟩= 0≥ lim sup
ε→0⟨u∗ε ,uε −u⟩
showing that limε→0 ⟨u∗ε ,uε −u⟩= 0. Thus
⟨u∗,u− v⟩ ≥ lim infε→0
(⟨u∗ε ,u−uε⟩+ ⟨u∗ε ,uε − v⟩)
= lim infε→0⟨u∗ε ,uε − v⟩ ≥ ⟨u∗ (v) ,u− v⟩
This implies u∗ ∈ A(u,ω) because if not, then by separation theorems, there exists v ∈ Vsuch that for all w∗ ∈ A(u,ω) ,
⟨u∗,u− v⟩< ⟨w∗,u− v⟩
contrary to what was shown above. Thus this obtains
Bu(t)−Bu0 +∫ t
0u∗ds =
∫ t
0f ds
where u∗ ∈ A(u,ω) . In case Buε (T ) ̸= B(uε (T )) , you do the same argument for T̂ < Twhere Buε
(T̂)= B
(uε
(T̂))
for all ε and for u. Then the above argument shows that
u∗X[0,T̂ ] ∈ A(X[0,T̂ ]u,ω
). This being true for every such T̂ < T implies that it holds on
[0,T ] and shows part of the following theorem which is the main result.