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Second-order subgradients of convex integral functionals

Authors:	Mohammed Moussaoui Alberto Seeger

Institution:	Department of Mathematics, University of Avignon, 33, rue Louis Pasteur, 84000 Avignon, France Alberto Seeger ; Department of Mathematics, University of Avignon, 33, rue Louis Pasteur, 84000 Avignon, France

Abstract:	The purpose of this work is twofold: on the one hand, we study the second-order behaviour of a nonsmooth convex function defined over a reflexive Banach space . We establish several equivalent characterizations of the set $\partial^2F(\overline x,\overline y)$ , known as the second-order subdifferential of at $\overline x$ relative to $\overline y\in \partial F(\overline x)$ . On the other hand, we examine the case in which is the functional integral associated to a normal convex integrand . 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 $(1\le p<+\infty)$ . We also establish a formula for computing the second-order subdifferential $\partial ^2I_f(\overline x,\overline y)$ .

Keywords:	Convex integral functional subdifferential second-order subdifferential Mosco convergence

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