Kolmogorov equation associated to the stochastic reflection problem ...

which follows taking into account (4.16).
Therefore there exists a sequence {ε k } such that
ϕ εk → ϕ, strongly in L 2 (K, ν)
Dϕ εk → Dϕ, weakly in L 2 (K, ν; H)
∫
∫
lim |Dϕ εk | 2 dν = |Dϕ| 2 dν.
k→∞
K
K
This implies that Dϕ εk → Dϕ strongly in L 2 (K, ν; H).
We finally assume that f ∈ L 2 (H, ν). Since Cb 1(H) is dense in L2 (K, ν),
there exists a sequence {f n } ⊂ Cb 1(H) strongly convergent in L2 (K; ν) to f.
Set ϕ n,ε = (λI − N ε ) −1 f n . By (4.12) we have
∫
|Dϕ ε − Dϕ n,ε | 2 dν ε ≤ 2 ∫
|f − f n | 2 dν,
λ
which implies
∫
H
K
|Dϕ ε − Dϕ n,ε | 2 dν ≤ 2 λ
So, again Dϕ εk → Dϕ strongly in L 2 (K, ν; H).
Step 3. We have
∫
K
K
|f − f n | 2 dν.
ϕ ∈ W 1,2
A (K, ν; H) ∩ W 2,2 (K; ν). (4.22)
By estimate (4.13) we have that {ϕ ε } is bounded in W 2,2 (K, ν). Therefore
there is a subsequence, still denoted {ϕ ε } which converges to ϕ in W 2,2 (K, ν).
In the same way we see that ϕ ∈ W 1,2
A
(K, ν; H).
Step 4. Checking the Neumann condition for ϕ.
From (4.14) we get
∫
N ε ϕ ε ψ dν = − 1 ∫
〈Dϕ ε , Dψ〉dν + 1
K 2
µ(K)
K
∫
Σ
ψ〈γ(Dϕ ε ), n(y)〉dµ Σ .
Recalling that N ε ϕ ε = λϕ ε − f −→ λϕ − f = Nϕ in L 2 (K, ν) and that
〈γ(Dϕ ε ), n(y)〉 → 〈γ(Dϕ), n(y)〉 in L 2 (Σ, µ Σ ) by Proposition 2.9, by (i) and
by (3.4) we obtain
∫
〈γ(Dϕ), n(y)〉 ψ dµ Σ = 0, ∀ψ ∈ W 1,2 (K, ν)
Σ
which implies (4.17) as claimed. This completes the proof.
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