Partial subdifferentials, derivates and Rademacher’s Theorem

Bessis, D.; Clarke, F.

doi:10.1090/S0002-9947-99-02203-5

Partial subdifferentials, derivates and Rademacher’s Theorem
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by D. N. Bessis and F. H. Clarke

Trans. Amer. Math. Soc. 351 (1999), 2899-2926

DOI: https://doi.org/10.1090/S0002-9947-99-02203-5

Published electronically: March 10, 1999

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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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