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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Partial subdifferentials, derivates and Rademacher's Theorem

Authors: D. N. Bessis and F. H. Clarke
Journal: Trans. Amer. Math. Soc. 351 (1999), 2899-2926
MSC (1991): Primary 26E99; Secondary 46G05, 49J50, 58B10.
Published electronically: March 10, 1999
MathSciNet review: 1475676
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Abstract: In this paper, we present new partial subdifferentiation formulas in nonsmooth analysis, based upon the study of two directional derivatives. Simple applications of these formulas include a new elementary proof of Rademacher's Theorem in ${\mathbb R}^n$, as well as some results on Gâteaux and Fréchet differentiability for locally Lipschitz functions in a separable Hilbert space.

RÉSUMÉ. Dans cet article, nous présentons de nouvelles formules de sousdifférentiation partielle en analyse nonlisse, basées sur l'étude de deux dérivées directionnelles. Une simple application de ces formules nous permet d'obtenir une nouvelle preuve élémentaire du théorème de Rademacher dans ${\mathbb R}^{n}$, ainsi que certains résultats sur la différentiabilité Gâteaux ou Fréchet des fonctions localement Lipschitz sur un espace de Hilbert séparable.

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Additional Information

D. N. Bessis
Affiliation: Centre for Process Systems Engineering, Imperial College of Science, Technology and Medicine, Exhibition Road, London, United Kingdom, SW7 2AZ

F. H. Clarke
Affiliation: Mathématiques, Université de Lyon I, 69622 Villeurbanne, France, and Centre de Recherches Mathématiques, Université de Montréal, C. P. 6128, Succ. Centre-ville, Montréal, Québec, Canada, H3C 3J7

Keywords: Nonsmooth analysis, locally Lipschitz functions, directional derivates, partial subdifferentials, G\^{a}teaux and Fr\'{e}chet derivatives.
Received by editor(s): February 2, 1997
Published electronically: March 10, 1999
Additional Notes: We gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada, and of le Fonds FCAR du Québec
Article copyright: © Copyright 1999 American Mathematical Society