1196 CHAPTER 34. GELFAND TRIPLES AND RELATED STUFF

Proposition 34.4.8 Let

X ={

u ∈ Lp (0,T ;V )≡ V : Lu≡ (Bu)′ ∈ Lp′ (0,T,V ′)}where V is a reflexive Banach space. Let a norm on X be given by

∥u∥X ≡ ∥u∥V +∥Lu∥V ′

Then there is a continuous function t → ⟨Bu,v⟩(t) such that ⟨Bu,v⟩(t) = ⟨B(u(t)) ,v(t)⟩a.e. t such that

supt∈[0,T ]

|⟨Bu,v⟩(t)| ≤C∥u∥X ∥v∥X

and if K : X → X ′

⟨Ku,v⟩ ≡∫ T

0⟨Lu,v⟩ds+ ⟨Bu,v⟩(0)

Then K is continuous and linear and

⟨Ku,u⟩= 12[⟨Bu,u⟩(T )+ ⟨Bu,u⟩(0)]

If u ∈ X and Bu(0) = 0 then there exists a sequence {un} such that ∥un−u∥X → 0 butun (t) = 0 for all t close to 0.

Proof: It only remains to verify the last assertion. Let ψn be increasing and piecewiselinear such that ψn (t) = 1 for t ≥ 2/n and equals 0 on [0,1/n]. Then clearly ψnu→ u inV .

(B(ψnu))′ = ψ′nBu+ψn (Bu)′

The second term converges to (Bu)′ in V ′. It remains to consider the first term.

∫ T

0

∥∥ψ′nBu∥∥p′

V ′ dt ≤∫ 2/n

0n∥∥∥∥∫ t

0(Bu)′ ds

∥∥∥∥p′

dt

≤ n∫ 2/n

0t p′−1

∫ t

0

∥∥(Bu)′∥∥p′

V ′ dsdt ≤∫ 2/n

0

∥∥(Bu)′∥∥p′

V ′ ds1p′(2/n)p′ n

Since p′ > 1, this converges to 0.Note that, by convolving with a mollifier, we could assume each un is also smooth.

In addition to this, we can draw a similar conclusion at the right endpoint. That is, ifBu(T ) = 0 there is a sequence {un} ⊆ X where un (t) = 0 for t near T which converges tou in X .

34.5 The Implicit Case, B = B(t)The above theorem can be generalized to the case where the formula is of the form

BX (t) = BX0 +∫ t

0Y (s)ds