2414 CHAPTER 70. MEASURABILITY WITHOUT UNIQUENESS

In this paper, we assume the following on p and F . The functions p and F are increasingand

δ2r−K ≤ p(r)≤ K(1+ r), r ≥ 0, (70.5.22)

p(r) = 0, r < 0,

F(r)≤ K(1+ r) r ≥ 0, (70.5.23)

F(r) = 0 if r < 0,

|µ (r1)−µ (r2)| ≤ Lip(µ) |r1− r2| , ||µ||∞ ≤C, (70.5.24)

and for a = F, p,and r1,r2 ≥ 0,

|a(r1)−a(r2)| ≤ K|r1− r2|. (70.5.25)

One can consider more general growth conditions than this, but we are keeping this partsimple to emphasize the new stochastic considerations.

It will be assumed thatσ i j = Ai jkluk,l +Ci jkl u̇k,l , (70.5.26)

where A and C are in L∞(U) and for B = A or C, we have the following symmetries.

Bi jkl = Bi jlk , B jikl = Bi jkl , Bi jkl = Bkli j , (70.5.27)

and we also assume for B = A or C that

Bi jklHi jHkl ≥ εHrsHrs (70.5.28)

for all symmetric H.Throughout the paper, V will be a closed subspace of (H1(U))3 containing the test

functions(C∞0 (U))3, ⇀ will denote weak or weak ∗ convergence while→ will mean strong

convergence. γ will denote the trace map from W 12(U) into L2(∂U). H will denote (L2(U))3 and we will always identify H and H ′ to write

V ⊆ H = H ′ ⊆V ′

We defineV = L2 (0,T ;V ) ,H = L2 (0,T,H) ,V ′ = L2 (0,T ;V ′

)70.5.1 The Abstract ProblemWe shall use two theorems found in Lions [91], and Simon [117] respectively. Thesetheorems apply for fixed ω . Proofs of generalizations of these theorems begin on Page2385.

Theorem 70.5.1 If p≥ 1 , q > 1 ,and W ⊆U ⊆ Y where the inclusion map of W into U iscompact and the inclusion map of U into Y is continuous, let

S = {u ∈ Lp(0,T ;W ) : u′ ∈ Lq(0,T ;Y ) and

||u||Lp(0,T ;W )+ ||u′||Lq(0,T ;Y ) < R}Then S is pre compact in Lp(0,T ;U).