2666 CHAPTER 78. A DIFFERENT APPROACH

Lemma 78.2.1 Let f be measurable into V ′ and let A satisfy the conditions 78.1.1 - 78.1.1.Then for K and M defined as above, it follows there exist measurable uε and w∗ε satisfying78.2.14.

Note this implies that, suppressing dependence on ω,

⟨Buε ,v⟩(0) = ⟨Bv(0) ,u0⟩

for all v ∈ X . Thus, letting v be a smooth function with values in V

⟨Buε (0) ,v(0)⟩= ⟨Bu0,v(0)⟩

Since V is dense in W, this requires Buε (0) = Bu0.Now define Λ to be the restriction of L to those u ∈ X which have Bu(0) = 0. Thus by

Corollary 78.1.9,

D(Λ) = {u ∈ X : Bu(0) = 0}= {u ∈ X : ⟨Bu,u⟩(0) = 0}

and if v ∈ D(Λ) ,u ∈ X , then as noted earlier,

⟨Ku,v⟩=∫ T

0⟨Lu,v⟩ds

Also, one can show an estimate for Λ∗.You can define D(T )≡ {u ∈ V : u′ ∈ V , u(T ) = 0} and let Tu =−Bu′. Then

⟨Tu,u⟩ = −∫ T

0

⟨Bu′,u

⟩=−⟨Bu,u⟩ |T0 +

∫ T

0

⟨(Bu)′ ,u

⟩= ⟨Bu,u⟩(0)+

∫ T

0

⟨B′u,u

⟩+∫ T

0

⟨Bu′,u

⟩and so we obtain

2⟨Tu,u⟩ ≥∫ T

0

⟨B′u,u

⟩(78.2.15)

Then one shows that T ∗ = Λ and that the graph of Λ∗ is the closure of the graph of T thusshowing that Λ∗ also satisfies an inequality like 78.2.15 for u ∈ D(Λ∗).

From 78.2.14,

ε⟨Lv,J−1Luε

⟩V ′,V + ⟨Kuε (ω) ,v⟩X ′,X + ⟨w∗ε (ω) ,v⟩V ′,V

= ⟨ f (ω) ,v⟩V ′,V + ⟨g(ω) ,v⟩V ′,V

If we restrict to v ∈ D(Λ) so Bv(0) = 0, then it reduces to

ε⟨Λv,J−1Luε

⟩V ′,V + ⟨Luε (ω) ,v⟩V ′,V + ⟨w∗ε (ω) ,v⟩V ′,V = ⟨ f (ω) ,v⟩V ′,V

and so J−1Luε ∈ D(Λ∗) . Thus, since D(Λ) is dense in V , it follows that

εΛ∗J−1Luε +Luε +w∗ε = f in V ′