77.3. PRELIMINARY RESULTS 2613

The following has to do with the values of Bu and gives an integration by parts formula.

Corollary 77.3.5 Let V ⊆W,W ′ ⊆ V ′ be separable Banach spaces, and B ∈ L (W,W ′)is nonnegative and self adjoint. Also suppose t → B(u(t)) has a weak derivative (Bu)′ ∈Lp′ (0,T,V ′) for u ∈ Lp (0,T,V ). Then there is a continuous function denoted as t→ Bu(t)which equals B(u(t)) a.e. t. Say for t /∈ N. Suppose Bu(0) = Bu0, u0 ∈W. Then

Bu(t) = Bu0 +∫ t

0(Bu)′ (s)ds in V ′ (77.3.7)

Then t→ Bu(t) is in C(NC,W ′

)and also for such t,

12⟨Bu(t) ,u(t)⟩= 1

2⟨Bu0,u0⟩+

∫ t

0

⟨(Bu)′ (s) ,u(s)

⟩ds

There exists a continuous function t → ⟨Bu,u⟩(t) which equals the right side of the abovefor all t and equals ⟨B(u(t)) ,u(t)⟩ off N. This also satisfies

supt∈[0,T ]

⟨Bu,u⟩(t)≤C(∥∥(Bu)′

∥∥Lp′ (0,T,V ′) ,∥u∥Lp(0,T,V )

)This also makes it easy to verify continuity of pointwise evaluation of Bu. Let Lu =

(Bu)′ .

u ∈ D(L)≡ X ≡{

u ∈ Lp (0,T,V ) : Lu≡ (Bu)′ ∈ Lp′ (0,T,V ′)}∥u∥X ≡max

(∥u∥Lp(0,T,V ) ,∥Lu∥Lp′ (0,T,V ′)

)(77.3.8)

Since L is closed, this X is a Banach space.Then the following theorem is obtained.

Theorem 77.3.6 Say (Bu)′ ∈ Lp′ (0,T,V ′) so

Bu(t) = Bu(0)+∫ t

0(Bu)′ (s)ds in V ′

the map u→ Bu(t) is continuous as a map from X to V ′. Also, if Y denotes those f ∈Lp ([0,T ] ,V ) for which f ′ ∈ Lp ([0,T ] ,V ) , so that f has a representative such that f (t) =f (0)+

∫ t0 f ′ (s)ds, then if ∥ f∥Y ≡ ∥ f∥Lp([0,T ],V )+∥ f ′∥Lp([0,T ],V ) , the map f → f (t) is con-

tinuous.

Also one can obtain the following for p > 1.

Proposition 77.3.7 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 ≡max(∥u∥V ,∥Lu∥V ′)