78.3. RELAXED COERCIVITY CONDITION 2675

From what was shown above, ⟨Buε ,uε⟩(0) = ⟨Bu0,u0⟩. Now passing to the limit asε → 0,

Lu+u∗ = f

in U ′r . But every term is in V ′ except the first and so it is also in V ′. Also, we know that

⟨Buε ,uε⟩(0) = ⟨Bu0,u0⟩ and by Theorem 78.1.8, ⟨Bu,u⟩(0) = ⟨Bu0,u0⟩ also. Then theintegration by parts formula yields

12⟨Bu,u⟩

(T̂)− 1

2⟨Bu0,u0⟩+

∫ T̂

0⟨u∗,u⟩dt =

∫ T̂

0⟨ f ,u⟩dt

which shows ∫ T̂

0⟨u∗,u⟩dt =

∫ T̂

0⟨ f ,u⟩dt− 1

2⟨Bu,u⟩

(T̂)+

12⟨Bu0,u0⟩

Then from 78.3.30 and the lower semicontinuity shown in 78.3.29, it follows that

lim supε→0

∫ T̂

0⟨u∗ε ,uε⟩ ≤

∫ T̂

0⟨ f ,u⟩dt +

12⟨Bu0,u0⟩− lim inf

ε→0

12⟨Buε ,uε⟩

(T̂)

≤∫ T̂

0⟨ f ,u⟩dt +

12⟨Bu0,u0⟩−

12⟨Bu,u⟩

(T̂)=∫ T̂

0⟨u∗,u⟩dt

Thus we have uε → u weakly in VI and (Buε)′→ (Bu)′ weakly in U ′

rI ,

lim supε→0

∫ T̂

0⟨u∗ε ,uε −u⟩ ≤

∫ T̂

0⟨u∗,u⟩−

∫ T̂

0⟨u∗,u⟩= 0

Therefore, by the limit condition 78.3, for any v ∈ V

lim infε→0

∫ T̂

0⟨u∗ε ,uε − v⟩ ≥

∫ T̂

0⟨u∗ (v) ,u− v⟩ , some u∗ (v) ∈ A(u,ω)

In particular, this holds for u and so, in fact,∫ T̂

0 ⟨u∗ε ,uε −u⟩ converges to 0. Therefore,

∫ T̂

0⟨u∗,u− v⟩ = lim

ε→0

∫ T̂

0⟨u∗ε ,u− v⟩

≥ lim infε→0

(∫ T̂

0⟨u∗ε ,u−uε⟩+

∫ T̂

0⟨u∗ε ,uε − v⟩

)

≥∫ T̂

0⟨u∗ (v) ,u− v⟩ , some u∗ (v) ∈ A(u,ω)

since v is arbitrary, this shows from separation theorems that u∗ (ω)∈A(u(ω) ,ω) in V ′[0,T̂ ]

.

This has proved the following theorem in which a more general coercivity condition isused.