79.4. STOCHASTIC INCLUSIONS WITHOUT UNIQUENESS ?? 2719

Suppose to the contrary that there exists y such that

η = lim infn→∞⟨zn,un− y⟩V ′,V < ⟨z̃,u− y⟩V ′,V , (79.4.68)

for all z̃ ∈ Âu. Take a subsequence, denoted still with subscript n such that

η = limn→∞⟨zn,un− y⟩V ′,V

Note that this subsequence does not depend on (t,ω). Thus

limn→∞⟨zn,un− y⟩V ′,V < ⟨z̃,u− y⟩V ′,V for all z̃ ∈ Âu (79.4.69)

We will obtain a contradiction to this. In what follows, we continue to use the subsequencejust described which satisfies the above inequality 79.4.69.

The estimate 79.4.66 implies,

0≤ ⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩− ≤ c(t,ω)+C ||u(t,ω)||pV , (79.4.70)

where c is a function in L1 ([0,T ]×Ω). Thanks to 79.4.65,

lim infn→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩ ≥ 0, a.e.

and, therefore, the following pointwise limit exists,

limn→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩− = 0, a.e.

and so we may apply the dominated convergence theorem using 79.4.70 and conclude

limn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩−dtdP

=∫

Ω

∫ T

0limn→∞⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩−dtdP = 0

Now, it follows from 79.4.67 and the above equation, that

limn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩+dtdP

= limn→∞

∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩

+⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩−dtdP

= limn→∞⟨zn,un−u⟩V ′,V = 0.

Therefore, both ∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩+dtdP

and ∫Ω

∫ T

0⟨zn (t,ω) ,un (t,ω)−u(t,ω)⟩−dtdP