1406 CHAPTER 42. MAXIMAL MONOTONE OPERATORS, HILBERT SPACE

Lemma 42.9.1 Under the conditions, 42.9.57 - 42.9.59, φ : H × [0,T ]→ [0,∞] is lowersemicontinuous.

Proof: Let (xn, tn)→ (x, t) and let λ ≡ liminfn→∞ φ (tn,xn) . Is

φ (t,x)≤ λ?

It suffices to assume λ < ∞ and by taking a subsequence, xn ∈ D for all n and

φ (tn,xn)→ λ .

Thenlim inf

n→∞φ (tn,xn) = lim inf

n→∞[φ (tn,xn)−φ (t,xn)+φ (t,xn)] . (42.9.60)

Nowlim sup

n→∞

|φ (tn,xn)−φ (t,xn)| ≤

lim supn→∞

K(

φ (tn,xn)+ |xn|2 +1)|tn− t|= 0.

Therefore, from 42.9.60

λ = lim infn→∞

φ (tn,xn) = lim infn→∞

φ (t,xn)≥ φ (t,x)

because of the assumption that φ (t, ·) is lower semicontinuous. This proves the lemma.In all that follows [x] is an element of L2 (0,T ;H) . Thus [x] is the equivalence class

of measurable square integrable functions which equal x a.e. This seems a little fussy butsince the existence results are based on surjectivity theorems and the Hilbert space theyapply to is L2 (0,T ;H) , it seems best to emphasize the equivalence classes of functions byusing this notation, at least while proving theorems on existence and uniqueness.

Corollary 42.9.2 For [x] ∈ L2 (0,T ;H) , t→ φ (t,x(t)) is measurable.

Proof: This follows because, due to Lemma 42.9.1, φ is Borel measurable and so φ ◦ xis also measurable.

Now define the following function, Φ, on the Hilbert space, L2 (0,T ;H) .

Φ([x])≡{ ∫ T

0 φ (t,x(t))dt if x(t) ∈ D for all t for some x(·) ∈ [x]+∞ otherwise

(42.9.61)

Note that since the functions φ (t, ·) are proper, the top condition is equivalent to the condi-tion ∫ T

0φ (t,x(t))dt if x(t) ∈ D a.e. for all x(·) ∈ [x] .

Lemma 42.9.3 Φ is convex, nonnegative, and lower semicontinuous on L2 (0,T ;H) .