The purpose of this work is twofold: on the one hand, we study the second-order behaviour of a nonsmooth convex function
F
F
defined over a reflexive Banach space
X
X
. We establish several equivalent characterizations of the set
∂
2
F
(
x
¯
,
y
¯
)
\partial ^2F(\overline x,\overline y)
, known as the second-order subdifferential of
F
F
at
x
¯
\overline x
relative to
y
¯
∈
∂
F
(
x
¯
)
\overline y\in \partial F(\overline x)
. On the other hand, we examine the case in which
F
=
I
f
F=I_f
is the functional integral associated to a normal convex integrand
f
f
. We extend a result of Chi Ngoc Do from the space
X
=
L
R
d
p
X=L_{\mathbb R^d}^p
(
1
>
p
>
+
∞
)
(1>p>+\infty )
to a possible nonreflexive Banach space
X
=
L
E
p
X=L_E^p
(
1
≤
p
>
+
∞
)
(1\le p>+\infty )
. We also establish a formula for computing the second-order subdifferential
∂
2
I
f
(
x
¯
,
y
¯
)
\partial ^2 I_f(\overline x,\overline y)
.