2672 CHAPTER 78. A DIFFERENT APPROACH

A sufficient condition for this condition is that ω → A(u(·,ω) ,ω) has a measurableselection into V ′ for any ω → u(·,ω) measurable into V and if u∗ ∈ A(u(·,ω) ,ω) ,then XIu∗ ∈ A(XIu(·,ω) ,ω) , and this is typical of what we will always consider,in which the values of u∗ are dependent on the earlier values of u only.

Let F be the duality map for r > max(p̂,2). Thus

⟨Fu,u⟩= ||u||r , ||Fu||= ||u||r−1

and is a demicontinuous map. Let X be those u∈Ur such that (Bu)′ ∈U ′r with a convenient

norm given by max(||u||Ur

,∣∣∣∣(Bu)′

∣∣∣∣U ′r

). Then if we let UrI play the role of VI in Theorem

78.2.2, we obtain the following lemma as a corollary of this theorem.

Lemma 78.3.1 Let A satisfy 78.3-78.3 and let f be measurable into V ′ and let u0 bemeasurable into W. Then for ε > 0, there exists a solution to

Lu+ εFu+u∗ = f , Bu(0,ω) = Bu0 (ω) (78.3.25)

such that Lu,u∗,u are all measurable into U ′r ,U

′r , and Ur respectively, u∗ (ω) ∈ A(u,ω).

In other terms, for v ∈ X = {u ∈Ur : Lu ∈U ′r }∫ T

0⟨Lu,v⟩+ ε

∫ T

0⟨Fu,v⟩+

∫ T

0⟨u∗,v⟩+

⟨Bu,v⟩(0) =∫ T

0⟨ f ,v⟩+ ⟨Bv(0) ,u0⟩ (78.3.26)

Proof: Using easy estimates and the definition that r > max(p̂,2) , p̂≥ p, (Recall thatp̂ determined the polynomial growth of ∥u∗∥V ′ where u∗ ∈ A(u,ω)) it is routine to showthat the earlier coercivity condition holds for εF +A(·,ω). Indeed, we have the followingfrom the above assumptions.

inf(∫ t

0⟨u∗,u⟩+λ (ω)⟨Bu,u⟩ds : u∗ ∈ A(u,ω)

)≥ δ (ω)

∫ t

0∥u∥p

V ds−m(ω)

Thus,

inf(∫ t

0⟨u∗,u⟩ : u∗ ∈ A(u,ω)

)≥ δ (ω)

∫ t

0∥u∥p

V ds

−CBλ (ω)∫ t

0η

(1η

)||u||2V −m(ω)

Then the right side is no smaller than

−CBλ (ω)∫ t

0

(1η

)η ||u||2V −m(ω)

≥ −CB,λ (ω)ηr/2∫ t

0||u||rU −CBλ (ω)T

(1η

)r/(r−2)

−m(ω)