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 derivative(2) F((x) over bar;(y) over bar), known as the second-order subdifferential of F at (x) over bar relative to (y) over bar is an element of partial derivative F((x) over bar). 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-Rd(p) (1 < p < +infinity) to a possible nonreflexive Banach space X = L-E(p) (1 less than or equal to p < +infinity). We also establish a formula for computing the second-order subdifferential partial derivative(2) I-f ((x) over bar, (y) over bar).