48.6. AN EXAMPLE 1573

≥∫

Ω

σ (un, t)∇u · (∇un−∇u)+∫

Ω

σ (un, t)∇un · (∇u−∇v)

The second term in the above converges to∫

Ωσ (u, t)∇u · (∇u−∇v).

Consider the first term after ≥ . It equals∫Ω

(σ (un, t)−σ (u, t))∇u · (∇un−∇u)+∫

Ω

σ (u, t)∇u · (∇un−∇u) (*)

The second of these terms converges to 0 because of weak convergence of un to u. As tothe first, if the measure of E is small enough, then(∫

E|∇u|2

)1/2

< δ

By Egoroff’s theorem, there is a set E having measure this small such that off this set,σ (un (x) , t)−σ (u(x) , t)→ 0 uniformly for x /∈E. Thus an application of Holder’s inequal-ity shows that |

∫EC (σ (un, t)−σ (u, t))∇u · (∇un−∇u)| ≤ δ whenever n is sufficiently

large thanks to the weak convergence of un to u which implies that ∇un−∇u is boundedin L2 (Ω)3. As to the integral over E, the fact that σ is bounded for fixed t implies theexistence of a constant C independent of n such that∣∣∣∣∫

Ω

(σ (un, t)−σ (u, t))∇uXE · (∇un−∇u)∣∣∣∣≤C

(∫E|∇u|2

)1/2

<Cδ

Thus the first term in * has absolute value no larger than (C+1)δ provided n is sufficientlylarge. Since δ is arbitrary, the limit of this term is 0. Thus,

lim infn→∞

∫Ω

σ (un, t)∇un · (∇un−∇v)≥∫

Ω

σ (u, t)∇u · (∇u−∇v)

This is a contradiction. Thus the liminf condition must hold.Next consider another operator. Let Σ1 be ∂Ω⧹ Σ0 and has positive surface measure.

Let r → a(r, t) be lower semicontinuous and r → b(r, t) be upper semicontinuous. Let0 < δ (t) ≤ a(r, t) ≤ b(r, t) ≤ 1

δ (t) . Also let both of these functions be measurable in t.

Now γ : V → L2 (Σ1) and so γ∗ : L2 (Σ1)→ V ′ defined in the usual way. Then z ∈ B(u, t)will mean z = γ∗w for some w ∈ L2 (Σ1) with

w(x) ∈ [a(γu(x) , t) ,b(γu(x) , t)]

for a.e. x such that⟨z,v⟩=

∫Σ1

w(x)γv(x)

Using Sobolev embedding theorems, if un→ u weakly in V , then from the Sobolev em-bedding theorem un→ u strongly in a suitable Sobolev space of fractional order such thatthe embedding of V into this space is compact and the trace map is still continuous. Thusthere is a subsequence such that γun (x)→ γu(x) pointwise a.e. and wn → w in L2 (Σ1).