is the equivalence class
of measurable square integrable functions which equal x a.e. This seems a little fussy but
since the existence results are based on surjectivity theorems and the Hilbert space they
apply to is L^{2}
(0,T;H )
, it seems best to emphasize the equivalence classes of
functions by using this notation, at least while proving theorems on existence and
uniqueness.
Corollary 33.7.2For
[x]
∈ L^{2}
(0,T;H )
,t → ϕ
(t,x (t))
is measurable.
Proof:This follows because, due to Lemma 33.7.1, ϕ is Borel measurable and so
ϕ ∘ x is also measurable.
Now define the following function, Φ, on the Hilbert space, L^{2}
(0,T;H )
.
{ ∫T
Φ([x]) ≡ 0 ϕ(t,x(t))dt if x (t) ∈ D for all t for some x(⋅) ∈ [x] (33.7.60)
+∞ otherwise
(33.7.60)
Note that since the functions ϕ
(t,⋅)
are proper, the top condition is equivalent to the
condition
∫ T
ϕ(t,x(t))dt if x (t) ∈ D a.e.for all x(⋅) ∈ [x].
0
Lemma 33.7.3 Φ is convex, nonnegative, and lower semicontinuous onL^{2}
(0,T ;H )
.
Proof:Since each ϕ
(t,⋅)
is nonnegative and convex, it follows that Φ is also
nonnegative and convex. It remains to verify lower semicontinuity. Suppose,
[xn]
→
[x]
in
L^{2}
(0,T ;H )
and let
λ = lim inf Φ ([xn]).
n→∞
Is λ ≥ Φ
([x])
? It suffices to assume λ < ∞, x_{n}
(t)
∈ D for all t, and x_{n}
(t)
→ x
(t)
a.e. say
for t
∈∕
N where N has measure zero. Let
{ x (t) if t ∕∈ N
^x (t) = x (t) if t ∈ N
1
Then
[^x]
=
[x]
and
^x
(t)
∈ D for all t. Then by pointwise convergence and Fatou’s
lemma,
∫ ∫
Φ([x]) = Φ([^x]) = T ϕ(t,^x(t))dt ≤ T lim inf ϕ (t,x (t))dt
0 0 n→∞ n
is bounded and so the first term in 33.7.66 is
dominated by an expression of the form Cλ. Now consider the middle term of 33.7.66.
Since ϕ is nonnegative,
∫ ∫ ∫ ∫
T 1-−tλ t sλ T T 1-−(t−λs)
0 λe 0 e ϕ(s,z(s))dsdt = 0 s λe dtϕ (s,z(s))ds
∫ T ∫ ∞ −u
≤ e duϕ(s,z(s))ds = Φ ([z]). (33.7.68)
0 0