Second-order subgradients of convex integral functionals

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