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

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

1. Aronszajn N., Differentiability of Lipschitzian mappings between Banach spaces, Studia Math. 57 (1976), 147-190. MR 54:13562
2. Bessis D.N., Analyse contingentielle et sousdifférentielle, PhD thesis, Université de Montréal, 1997.
3. Borwein J.M., Strójwas H.M., Proximal analysis and boundaries of closed sets in Banach space. II. Applications, Canad. J. Math. 39 (1987), no. 2, 428-472. MR 88f:46034
4. Bruckner A., Differentiation of real functions, Second edition, CRM Monograph Series, 1994. MR 94m:26001
5. Choquet G., Outils topologiques et métriques de l'analyse mathématique, Cours rédigé par C. Mayer, Centre de Documentation Universitaire, Paris, 1969. MR 41:7033
6. Clarke F.H., Generalized gradients and applications, Trans. Amer. Math. Soc. 205 (1975), 247-262. MR 51:3373
7. Clarke F.H., Optimization and nonsmooth analysis, Second edition, Classics in Applied Mathematics 5, Society for Industrial and Applied Mathematics (SIAM), Philadelphia, 1990. MR 91e:49001
8. Clarke F.H., Ledyaev Yu.S., Stern R.J., Wolenski P.R., Nonsmooth analysis and optimal control, to be published in Graduate Texts in Mathematics, Springer-Verlag. CMP 98:06
9. Clarke F.H., Ledyaev Yu.S., Wolenski P.R., Proximal analysis and minimization principles, J. Math. Anal. Appl. 196 (1995), 722-735. MR 96i:49028
10. Dol\v{z}enko E.P., Boundary properties of arbitrary functions (Russian), Izv. Akad. Nauk SSSR Ser. Mat. 31 (1967), 3-14. MR 36:388
11. Fabián M., Preiss D., On intermediate differentiability of Lipschitz functions on certain Banach spaces, Proc. Amer. Math. Soc. 113 (1991), no. 3, 733-740. MR 92g:46048
12. Halmos P.R., Measure Theory, D.Van Nostrand Company, Inc., New York, 1986. MR 11:504d
13. Giles J.R., Sciffer S., Locally Lipschitz functions are generically pseudo-regular on separable Banach spaces, Bull. Austral. Math. Soc. 47 (1993), 205-212. MR 94a:58022
14. Kuratowski K., Topology, New edition, revised and augmented, Academic Press, New York-London, 1966. MR 36:840
15. Luke\v{s} J., Malý J., Zajicek L., Fine topology methods in real analysis and potential theory, Lecture Notes in Mathematics, 1189, Springer-Verlag, Berlin-New York, 1986. MR 89b:31001
16. Mignot F., Contrôle dans les inéquations variationnelles elliptiques, J. Funct. Anal. 22 (1976), 130-185. MR 54:11136
17. de Mises M.R., La base géométrique du théorème de M. Mandelbrojt sur les points singuliers d'une fonction analytique, C.R. Acad. Sci. Paris Sér. I Math. 205 (1937), 1353-1355.
18. Nekvinda A., Zajicek L., A simple proof of Rademacher Theorem, \v{C}asopis P\v{e}st. Mat. 113 (1988), no. 4, 337-341. MR 90c:26043
19. Phelps R.R., Convex functions, monotone operators and differentiability, Second edition, Lecture Notes in Mathematics, 1364, Springer-Verlag, Berlin, 1993. MR 94f:46055
20. Phelps R.R., Gaussian null sets and differentiability of Lipschitz map on Banach spaces, Pacific J. Math. 77 (1978), no. 2, 523-531. MR 80m:46040
21. Preiss D., Differentiability of Lipschitz functions on Banach spaces, J. Funct. Anal. 91 (1990), 312-345. MR 91g:46051
22. Preiss D., Gâteaux differentiable Lipschitz functions need not be Fréchet differentiable on a residual subset, Rend. Circ. Mat. Palermo Suppl. 2 (1982), 217-222. MR 84h:46054
23. Rademacher H., Über partielle und totale Differenzierbarkeit I., Math. Ann. 89 (1919), 340-359.
24. Saint-Pierre J., Sur le théorème de Rademacher, Travaux Sém. Anal. Convexe 12 (1982), exp. no. 2, 2.1-2.10. MR 84f:26016
25. Saks S., Theory of the integral, Second revised edition, Dover Publications, Inc., New York, 1964. MR 29:4850
26. Shu Zhong S., Choquet theorem and nonsmooth analysis, J. Math. Pures Appl. 67 (1988), no. 4, 411-432. MR 90e:58008
27. Shu Zhong S., Différentiabilité d'une fonction localement lipschitzienne dans un espace de Banach séparable, J. Systems Sci. Math. Sci. 3 (1983), no. 2, 112-119. MR 88i:58013
28. Zajícek L., Differentiability of the distance function and points of multivaluedness of the metric projection in Banach space, Czechoslovak Math. J. 33(108) (1983), no. 2, 292-308. MR 85b:46025
29. Zajícek L., Sets of $\sigma$ -porosity and sets of $\sigma$ -porosity , Casopis Pest. Mat. 101 (1976), no. 4, 350-359. MR 56:15935