Critical values of set-valued maps with stratifiable graphs. Extensions of Sard and Smale-Sard theorems

Ioffe, A.

doi:10.1090/S0002-9939-08-09101-6

Critical values of set-valued maps with stratifiable graphs. Extensions of Sard and Smale-Sard theorems
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Proc. Amer. Math. Soc. 136 (2008), 3111-3119 Request permission

Abstract:

We prove three theorems extending Sard’s theorem and its infinite dimensional extension due to Smale to set-valued mappings with stratifiable graphs. The very concept of a critical value comes from (nonsmooth) variational analysis and turns out to be perfectly compatible with the natural condition defining “good” stratifications (e.g., Whitney stratification in the finite dimensional case).

References

Jean-Pierre Aubin and Ivar Ekeland, Applied nonlinear analysis, Pure and Applied Mathematics (New York), John Wiley & Sons, Inc., New York, 1984. A Wiley-Interscience Publication. MR 749753
Jérôme Bolte, Aris Daniilidis, Adrian Lewis, and Masahiro Shiota, Clarke subgradients of stratifiable functions, SIAM J. Optim. 18 (2007), no. 2, 556–572. MR 2338451, DOI 10.1137/060670080
M. Coste, An Introduction to $o$-Minimal Geometry, Inst. Rech. Math., Univ. de Rennes, 1999 (http://name.math.univ-rennes1.fr/michel.coste/polyens/OMIN.pdf).
Lou van den Dries, Tame topology and o-minimal structures, London Mathematical Society Lecture Note Series, vol. 248, Cambridge University Press, Cambridge, 1998. MR 1633348, DOI 10.1017/CBO9780511525919
Lou van den Dries and Chris Miller, Geometric categories and o-minimal structures, Duke Math. J. 84 (1996), no. 2, 497–540. MR 1404337, DOI 10.1215/S0012-7094-96-08416-1
Christopher G. Gibson, Klaus Wirthmüller, Andrew A. du Plessis, and Eduard J. N. Looijenga, Topological stability of smooth mappings, Lecture Notes in Mathematics, Vol. 552, Springer-Verlag, Berlin-New York, 1976. MR 0436203, DOI 10.1007/BFb0095244
A. D. Ioffe, Directional compactness, scalarization and nonsmooth semi-Fredholm mappings, Nonlinear Anal. 29 (1997), no. 2, 201–219. MR 1446225, DOI 10.1016/S0362-546X(96)00046-6
A. D. Ioffe, Metric regularity and subdifferential calculus, Uspekhi Mat. Nauk 55 (2000), no. 3(333), 103–162 (Russian, with Russian summary); English transl., Russian Math. Surveys 55 (2000), no. 3, 501–558. MR 1777352, DOI 10.1070/rm2000v055n03ABEH000292
A. Ioffe, A Sard theorem for tame set-valued mappings, J. Math. Anal. Appl. 335 (2007), no. 2, 882–901. MR 2345506, DOI 10.1016/j.jmaa.2007.01.104
Tosio Kato, Perturbation theory for linear operators, Classics in Mathematics, Springer-Verlag, Berlin, 1995. Reprint of the 1980 edition. MR 1335452, DOI 10.1007/978-3-642-66282-9
K. Kurdyka, P. Orro, and S. Simon, Semialgebraic Sard theorem for generalized critical values, J. Differential Geom. 56 (2000), no. 1, 67–92. MR 1863021, DOI 10.4310/jdg/1090347525
A. S. Lewis, Active sets, nonsmoothness, and sensitivity, SIAM J. Optim. 13 (2002), no. 3, 702–725 (2003). MR 1972212, DOI 10.1137/S1052623401387623
Boris S. Mordukhovich, Variational analysis and generalized differentiation. I, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 330, Springer-Verlag, Berlin, 2006. Basic theory. MR 2191744
R. Tyrrell Rockafellar and Roger J.-B. Wets, Variational analysis, Grundlehren der mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], vol. 317, Springer-Verlag, Berlin, 1998. MR 1491362, DOI 10.1007/978-3-642-02431-3
S. Smale, An infinite dimensional version of Sard’s theorem, Amer. J. Math. 87 (1965), 861–866. MR 185604, DOI 10.2307/2373250
Hassler Whitney, Local properties of analytic varieties, Differential and Combinatorial Topology (A Symposium in Honor of Marston Morse), Princeton Univ. Press, Princeton, N.J., 1965, pp. 205–244. MR 0188486