77.6. VARIATIONAL INEQUALITIES 2635
u∗n→ u∗ weakly in V ′
Pun→ ξ weakly in V ′
Let Λ denote those v ∈ V such that (Bv)′ ∈ V ′ and Bv(0) = 0. Then for v ∈ Λ,⟨(Bun)
′ ,un− v⟩+ ⟨u∗n (·,ω) ,un− v⟩+n⟨P(un (·,ω)) ,un− v⟩= ⟨ f (·,ω) ,un− v⟩
Thus by monotonicity considerations,⟨(Bv)′ ,un− v
⟩+ ⟨u∗n (·,ω) ,un− v⟩+n⟨P(un (·,ω)) ,un− v⟩ ≤ ⟨ f (·,ω) ,un− v⟩ (*)
It follows that
lim supn→∞
⟨P(un (·,ω)) ,un− v⟩ ≤ 0
lim supn→∞
⟨P(un (·,ω)) ,un−u⟩ ≤ ⟨−ξ ,u− v⟩
Now, since Λ is dense, v can be chosen as close as desired to u and hence
lim supn→∞
⟨P(un (·,ω)) ,un−u⟩ ≤ 0
Since P is monotone, in fact the limit exists in the above. Therefore, for any v ∈ Λ and ∗,
lim infn→∞
(⟨P(un (·,ω)) ,un− v⟩)≥ ⟨Pu,u− v⟩
and so⟨Pu,u− v⟩ ≤ 0
for all v ∈ Λ. It follows that Pu = 0 and so u ∈K .Now for v ∈ Λ∩K , monotonicity considerations imply⟨
(Bv)′ ,un− v⟩+ ⟨u∗n (·,ω) ,un−u⟩+ ⟨u∗n (·,ω) ,u− v⟩ ≤ ⟨ f (·,ω) ,un− v⟩
Then
⟨u∗n (·,ω) ,un−u⟩ ≤ ⟨ f (·,ω) ,un− v⟩−⟨(Bv)′ ,un− v
⟩−⟨u∗n (·,ω) ,u− v⟩ (77.6.65)
Then
lim supn→∞
⟨u∗n (·,ω) ,un−u⟩ ≤ ⟨ f (·,ω) ,u− v⟩+⟨(Bv)′ ,v−u
⟩+ ⟨u∗ (·,ω) ,v−u⟩
We assume the existence of a regularizing sequence. If u ∈K there exists ui→ u weaklyin V such that
lim supi→∞
⟨(Bui)
′ ,ui−u⟩V≤ 0
In the above inequality, let v = ui
lim supn→∞
⟨u∗n (·,ω) ,un−u⟩ ≤ ⟨ f (·,ω) ,u−ui⟩+⟨(Bui)
′ ,ui−u⟩+ ⟨u∗ (·,ω) ,ui−u⟩