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Transactions of the American Mathematical Society
Transactions of the American Mathematical Society
ISSN 1088-6850(online) ISSN 0002-9947(print)

 

Second-order subgradients of
convex integral functionals


Authors: Mohammed Moussaoui and Alberto Seeger
Journal: Trans. Amer. Math. Soc. 351 (1999), 3687-3711
MSC (1991): Primary 49J52, 28B20
Published electronically: March 1, 1999
MathSciNet review: 1487628
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Abstract: 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 $\partial ^2I_f(\overline x,\overline y)$.


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Additional Information

Mohammed Moussaoui
Affiliation: Department of Mathematics, University of Avignon, 33, rue Louis Pasteur, 84000 Avignon, France

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

DOI: http://dx.doi.org/10.1090/S0002-9947-99-02248-5
PII: S 0002-9947(99)02248-5
Keywords: Convex integral functional, subdifferential, second-order subdifferential, Mosco convergence.
Received by editor(s): June 10, 1996
Received by editor(s) in revised form: March 13, 1997
Published electronically: March 1, 1999
Article copyright: © Copyright 1999 American Mathematical Society