1572 CHAPTER 48. MULTIFUNCTIONS AND THEIR MEASURABILITY

48.6 An ExampleLet σ (r, t) be a continuous function of r which satisfies

r → σ (r, t) is continuous, t→ σ (r, t) is measurable,0 < δ (t)≤ σ (r, t)≤ 1/δ (t)

There is no uniform lower bound needed for δ (t). Then we will let V be a closed subspaceof H1 (Ω) where Ω is a bounded open set with Lipschitz boundary.

V ≡{

u ∈ H1 (ω) : γu = 0 on Σ0}

where α (Σ0) > 0 for α the surface measure, Σ0 a closed subset of ∂Ω and γ is the tracemap. Thus an equivalent norm for V is

∥u∥2V =

∫Ω

|∇u|2 dx

Also let H = L2 (Ω) and let H = H ′ so that V ⊆H = H ′ ⊆V ′. Then let A(·, t) : V →V ′ bedefined by

⟨A(u, t) ,v⟩ ≡∫

Ω

σ (u, t)∇u ·∇v

Is this a bounded pseudomonotone map? It is clearly bounded thanks to the bounds on σ .Suppose then that un→ u weakly in V and

lim supn→∞

⟨A(un, t) ,un−u⟩ ≤ 0

Does the liminf condition hold? If not, then there exists a subsequence and v ∈V such that

limn→∞⟨A(un, t) ,un− v⟩< ⟨A(u, t) ,u− v⟩

By compactness, there is a further subsequence still denoted with n such that un → ustrongly in L2 (Ω) and pointwise. Consider∫

Ω

σ (un, t)∇un · (∇un−∇v)

Now by the dominated convergence theorem,∫Ω

|σ (un, t)−σ (u, t)|2→ 0

and so in fact σ (un, t)∇un→ σ (u, t)∇u weakly in H3. Then∫Ω

σ (un, t)∇un · (∇un−∇v) =∫

Ω

σ (un, t)∇un · (∇un−∇u)

+∫

Ω

σ (un, t)∇un · (∇u−∇v)