78.3. RELAXED COERCIVITY CONDITION 2671

situations. If u ∈ V , then we will always consider u ∈ VI also by simply considering itsrestriction to I. With this convention, it is clear that if u is measurable into V then it is alsomeasurable into VI .

Then the modified conditions on A : VI →P (V ′I ) are as follows for A(u,ω) a convexclosed set in V ′I whenever u ∈ VI .

• growth estimateAssume the specific estimate for u ∈ VI .

sup{∥u∗∥V ′I : u∗ ∈ A(u,ω)

}≤ a(ω)+b(ω)∥u∥p̂−1

VI(78.3.19)

where a(ω) ,b(ω) are nonnegative, p̂≥ p.

• coercivity estimateAlso assume the coercivity condition: valid for each t ≤ T and for some λ (ω)≥ 0,

inf(∫ t

0⟨u∗,u⟩+λ (ω)⟨Bu,u⟩ds : u∗ ∈ A(u,ω)

)

≥ δ (ω)∫ t

0∥u∥p

V ds−m(ω) (78.3.20)

where m(ω) is some nonnegative constant for fixed ω, and δ (ω)> 0. No uniformityin ω is necessary.

• Limit conditionsLet U be a Banach space dense and compact in V and that if ui ⇀ u in VI andu∗i ∈ A(ui,ω) with (Bun)

′→ (Bu)′ weakly in U ′rI , then if

lim supi→∞

⟨u∗i ,ui−u⟩V ′I ,VI≤ 0 (78.3.21)

it follows that for all v ∈ VI , there exists u∗(v) ∈ Au such that

lim infi→∞⟨u∗i ,ui− v⟩V ′I ,VI

≥ ⟨u∗ (v) ,u− v⟩V ′I ,VI(78.3.22)

You typically obtain this kind of thing from Theorem 78.1.2 applied to lower orderterms along with some sort of compactness of the embedding of V into W .

• measurability conditionFor ω → u(·,ω) measurable into V ,

ω → A(XIu(·,ω) ,ω) has a measurable selection into V ′I . (78.3.23)

This condition means there is a function ω → u∗ (ω) which is measurable into V ′Isuch that u∗ (ω) ∈ A(XIu(·,ω) ,ω) . This is assured to take place if the followingstandard measurability condition is satisfied for all O open in V ′I :

{ω : A(XIu(·,ω) ,ω)∩O ̸= /0} ∈F (78.3.24)