2634 CHAPTER 77. STOCHASTIC INCLUSIONS

Now for fixed ω,qτr (t,ω) does not change for all r large enough. This is because it is acontinuous function of t and so is bounded on the interval [0,T ]. Thus, for r large enoughand fixed ω , qτr (t,ω) = q(t,ω) . Thus, we obtain

⟨Bwr (t,ω) ,wr (t,ω)⟩+∫ t

0∥wr (s,ω)∥p

V ds≤C (ω) (77.5.63)

Now, as before one can pass to a limit involving a subsequence, as r→ ∞ and obtain asolution to the integral equation

Bw(t,ω)−Bu0 (ω)+∫ t

0z(s,ω)ds =

∫ t

0f (s,ω)ds

where z(s,ω) ∈ A(s,ω,w(s,ω)+q(s,ω)) for a.e. s and z is product measurable. Thenan application of Theorem 77.2.10 shows that there exists a solution w to this integralequation for each ω which also has (t,ω)→ w(t,ω) product measurable and (t,ω)→z(t,ω) product measurable. Now just let u(t,ω) = w(t,ω)+q(t,ω) .

The last claim follows from letting t = a in the top equation and then subtracting thisfrom the top equation with t > a.

77.6 Variational InequalitiesWe have some good theorems above in the context of 77.5.25 - 77.5.28, 77.5.30 - 77.5.34and B satisfies 77.3.6 and assume, if it depends on ω, it is of the form

B(ω) = k (ω)B, k (ω)≥ 0, k measurable

Now this will be used to consider variational inequalities.Let K be a closed convex subset of V containing 0. Let P : V → V ′ be an operator of

penalization. Thus P = 0 on K and is monotone and demicontinuous and nonzero off K .

Pu = F (u−projK (u))

where F is the duality map such that ⟨Fu,u⟩ = ∥u∥2 ,∥Fu∥ = ∥u∥. Then A(·,ω) + nPsatisfies the conditions for Theorem 77.5.6 assuming A(·,ω) satisfies the conditions ofthis theorem. Then by Theorem 77.5.6, there exists a solution un such that (t,ω) →un (t,ω) ,(t,ω)→ u∗n (t,ω) are product measurable, and for each ω,

(Bun)′+u∗n (·,ω)+nP(un (·,ω)) = f (·,ω) in V ′

Bun (0,ω) = 0 (77.6.64)

Here B is as described in that theorem. Using 0 ∈K and monotonicity of P, the estimatesfor A lead to an estimate of the form

∥un (·,ω)∥V +∥u∗n (·,ω)∥V ′ ≤C (ω)

Then there is a subsequenceun→ u weakly in V