2718 CHAPTER 79. INCLUDING STOCHASTIC INTEGRALS

Now, the coercivity condition 3 shows that if y ∈ V , then

⟨zn (t,ω) ,un (t,ω)− y(t,ω)⟩ ≥ b3 ||un (t,ω)||pV −b4 (t,ω)−λ |un (t,ω)|2H−(

b1 ||un (t,ω)||p−1 +b2 (t,ω))||y(t,ω)||V .

Using p−1 = pp′ , where 1

p +1p′ = 1, the right-hand side of this inequality equals

b3 ||un (t,ω)||pV −b4 (t,ω)−b1 ||un (t,ω)||p/p′ ||y(t,ω)||V−b2 (t,ω) ||y(t,ω)||V −λ |un (t,ω)|2H ,

the last term being bounded by a function in L1 ([0,T ]×Ω) by assumption. Thus thereexists cy (·, ·) ∈ L1 ([0,T ]×Ω) and a positive constant C such that

⟨zn (t,ω) ,un (t,ω)− y(t,ω)⟩ ≥ −cy (t,ω)−C ||y(t,ω)||pV . (79.4.66)

Letting y = u, we use Fatou’s lemma to write

lim infn→∞

∫Ω

∫ T

0

(⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩+ cu (t,ω)+C ||u(t,ω)||pV

)dtdP≥

∫Ω

∫ T

0lim inf

n→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩+

(cu (t,ω)+C ||u(t,ω)||pV

)dtdP

≥∫

Ω

∫ T

0

(cu (t,ω)+C ||u(t,ω)||pV

)dtdP.

Here, we added the term cu (t,ω) +C ||u(t,ω)||pV to make the integrand nonnegative inorder to apply Fatou’s lemma. Thus,

lim infn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩dtdP≥ 0.

Consequently, using the claim in the last inequality,

0 ≥ lim supn→∞

⟨zn,un−u⟩V ′,V

≥ lim infn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩dtdP

≥∫

Ω

∫ T

0lim inf

n→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩dtdP≥ 0

hence, we find thatlimn→∞⟨zn,un−u⟩V ′,V = 0. (79.4.67)

We need to show that if y is given in V then

lim infn→∞⟨zn,un− y⟩V ′,V ≥ ⟨z(y) ,u− y⟩ V ′,V , z(y) ∈ Âu