77.5. THE MAIN RESULT 2627

Then for δ small enough, depending on ε,

p̂r

δr/p̂ <

ε

2

And so the inequality ending at 77.5.38 yields

⟨Buh,uh⟩(T )+ ε ∥uh∥rU ≤C

(∥ f∥p̂′

V ′ ,(a(ω)+1)T,ε)+ ⟨Bu0,u0⟩

From 77.5.37 and the boundedness of the various operators, (B(ω)uh (·,ω))′ is boundedin U ′. Thus, summarizing these estimates yields the following for fixed ε∥∥(B(ω)uh (·,ω))′

∥∥U ′ +∥uh∥U +∥u∗h∥V ′ ≤C (77.5.39)

where C does not depend on h although it does depend on ε and of course on ω . Then onecan get a subsequence, still denoted with h such that as h→ 0,

uh→ u weakly in U (77.5.40)

τhuh→ u weakly in U (77.5.41)

(B(ω)uh (·,ω))′→ (B(ω)u)′ weakly in U ′ (77.5.42)

uh→ u strongly in V (77.5.43)

u∗h→ u∗ weakly in V ′ (77.5.44)

Fuh→ ξ ∈U ′ (77.5.45)

Bu(0,ω) = Bu0 (ω) (77.5.46)

The fourth of these comes from a use of Theorem 77.3.1. We need to argue that u∗ ∈A(u,ω). From the equation and initial conditions of 77.5.37, it follows from monotonicityconditions and the observation that V ′ is contained in U ′ that⟨

(B(ω)uh (·,ω))′ ,uh−u⟩+ ⟨εFuh (·,ω) ,uh−u⟩

+⟨u∗h (·,ω) ,uh−u⟩= ⟨ f (·,ω) ,uh−u⟩

and so ⟨(B(ω)u(·,ω))′ ,uh−u

⟩+ ⟨εFuh (·,ω) ,uh−u⟩U ′,U

+⟨u∗h (·,ω) ,uh−u⟩V ′,V ≤ ⟨ f (·,ω) ,uh−u⟩

by the strong convergence of 77.5.43, it follows that the third term converges to 0 as h→ 0.This is because the estimate 77.5.27 implies that the u∗h are bounded, and then the strongconvergence gives the desired result. Hence

lim suph→0⟨εFuh (·,ω) ,uh−u⟩U ′,U ≤ 0

and since F is monotone and hemicontinuous, it follows that in fact,

limh→0⟨εFuh (·,ω) ,uh−u⟩U ′,U = 0