2544 CHAPTER 76. IMPLICIT STOCHASTIC EQUATIONS

Lemma 76.3.2 If p≥ 2 and

⟨A(t,u,ω) ,u⟩V ≥ δ ∥u∥pV − c(t,ω) (76.3.11)

∥A(t,u,ω)∥V ′ ≤ k∥u∥p−1V + c1/p′ (t,ω) (76.3.12)

where c≥ 0, c ∈ L1 ([0,T ]×Ω) , then if (t,ω)→ q(t,ω) is in V ,, it follows that for a.e. ω,similar inequalities hold for Ā given by

Ā(t,u,ω)≡ A(t,u+q(t,ω) ,ω) (76.3.13)

Proof: Letting q be progressively measurable, q(t,ω) ∈ V only consider ω such thatt→ q(t,ω) is in Lp (0,T,V ). ⟨

Ā(t,u,ω) ,u⟩=

⟨A(t,u+q(t,ω) ,ω) ,u⟩ = ⟨A(t,u+q(t,ω) ,ω) ,u+q(t,ω)⟩−⟨A(t,u+q(t,ω) ,ω) ,q(t,ω)⟩

≥ δ ∥u+q(t,ω)∥pV − k∥u+q(t,ω)∥p−1

V ∥q(t,ω)∥V − c1/p′ (t,ω)∥q(t,ω)∥V − c(t,ω)

≥ δ ∥u+q(t,ω)∥pV − k∥u+q(t,ω)∥p−1

V ∥q(t,ω)∥V −∥q(t,ω)∥pV −2c(t,ω)

≥ δ

2∥u+q(t,ω)∥p

V −C (k,δ ,T )∥q(t,ω)∥pV −2c(t,ω)

Now∥u+q(t,ω)∥ ≥ ∥u∥−∥q(t,ω)∥

and so by convexity,

∥u+q(t,ω)∥p +∥q(t,ω)∥p

2≥(∥u+q(t,ω)∥+∥q(t,ω)∥

2

)p

≥(∥u∥

2

)p

This implies

∥u+q(t,ω)∥p ≥ 2(∥u∥p

2p −∥q(t,ω)∥p

2

)Therefore, ⟨

Ā(t,u,ω) ,u⟩=

⟨A(t,u+q(t,ω) ,ω) ,u⟩ ≥ δ

2

(2(∥u∥p

2p −∥q(t,ω)∥p

2

))−C (k,δ ,T )∥q(t,ω)∥p

V −2c(t,ω)

≥ δ

2p ∥u∥p− c′ (t,ω)

where c′ ∈ L1 ([0,T ]×Ω).