Structure of level sets and Sard-type properties of Lipschitz maps
Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, p. 863-902

We consider certain properties of maps of class C 2 from d to d-1 that are strictly related to Sard’s theorem, and we show that some of them can be extended to Lipschitz maps, while others require some additional regularity. We also give examples showing that, in terms of regularity, our results are optimal.

Published online : 2019-02-21
Classification:  26B35,  26B10,  26B05,  49Q15,  58C25
@article{ASNSP_2013_5_12_4_863_0,
     author = {Alberti, Giovanni and Bianchini, Stefano and Crippa, Gianluca},
     title = {Structure of level sets and Sard-type properties of Lipschitz maps},
     journal = {Annali della Scuola Normale Superiore di Pisa - Classe di Scienze},
     publisher = {Scuola Normale Superiore, Pisa},
     volume = {Ser. 5, 12},
     number = {4},
     year = {2013},
     pages = {863-902},
     zbl = {1295.26016},
     mrnumber = {3184572},
     language = {en},
     url = {http://www.numdam.org/item/ASNSP_2013_5_12_4_863_0}
}
Alberti, Giovanni; Bianchini, Stefano; Crippa, Gianluca. Structure of level sets and Sard-type properties of Lipschitz maps. Annali della Scuola Normale Superiore di Pisa - Classe di Scienze, Serie 5, Volume 12 (2013) no. 4, pp. 863-902. http://www.numdam.org/item/ASNSP_2013_5_12_4_863_0/

[1] G. Alberti, S. Bianchini and G. Crippa, A uniqueness result for the continuity equation in two dimensions, J. Eur. Math. Soc. (JEMS), to appear. Available at http/cvgmt.sns.it/ | MR 3161282 | Zbl 1286.35006

[2] S. M. Bates, Toward a precise smoothness hypothesis in Sard’s theorem, Proc. Amer. Math. Soc. 117 (1993), 279–283. | MR 1112486 | Zbl 0767.58003

[3] B. Bojarski, P. Hajłasz and P. Strzelecki, Sard’s theorem for mappings in Hölder and Sobolev spaces, Manuscripta Math. 118 (2005), 383–397. | MR 2183045 | Zbl 1098.46024

[4] J. Bourgain, M. V. Korobkov and J. Kristensen, On the Morse-Sard property and level sets of Sobolev and BV functions, Rev. Mat. Iberoam. 29 (2013), 1–23. | MR 3010119 | Zbl 1273.26017

[5] L. De Pascale, The Morse-Sard theorem in Sobolev spaces, Indiana Univ. Math. J. 50 (2001), 1371–1386. | MR 1871360 | Zbl 1027.58008

[6] A.Ya. Dubovickiĭ, On the structure of level sets of differentiable mappings of an n-dimensional cube into a k-dimensional cube (Russian), Izv. Akad. Nauk SSSR. Ser. Mat. 21 (1957), 371–408. | MR 94424 | Zbl 0050.28102

[7] R. Engelking, “General Topology”, revised and completed edition, Sigma Series in Pure Mathematics, Vol. 6, Heldermann Verlag, Berlin, 1989. | MR 1039321 | Zbl 0684.54001

[8] K. J. Falconer, “The Geometry of Fractal Sets”, Cambridge Tracts in Mathematics, Vol. 85, Cambridge University Press, Cambridge, 1985. | MR 867284 | Zbl 0587.28004

[9] H. Federer, “Geometric Measure Theory”, Grundlehren der mathematischen Wissenschaften, Vol. 153, Springer, Berlin-New York 1969; reprinted in the series Classics in Mathematics, Springer, Berlin-Heidelberg, 1996. | MR 257325 | Zbl 0176.00801

[10] A. Figalli, A simple proof of the Morse-Sard theorem in Sobolev spaces, Proc. Amer. Math. Soc. 136 (2008), 3675–3681. | MR 2415054 | Zbl 1157.58002

[11] E. L. Grinberg, On the smoothness hypothesis in Sard’s theorem, Amer. Math. Monthly 92 (1985), 733–734. | MR 820057 | Zbl 0633.58002

[12] M. W. Hirsch, “Differential Topology”, corrected reprint of the 1976 original, Graduate Texts in Mathematics, Vol. 33, Springer-Verlag, New York, 1976. | MR 448362 | Zbl 0356.57001

[13] S. V. Konyagin, On the level sets of Lipschitz functions, Tatra Mt. Math. Publ. 2 (1993), 51–59. | MR 1251037 | Zbl 0790.26012

[14] S. G. Krantz and H. R. Parks, “Geometric Integration Theory”, Cornerstones, Birkhäuser, Boston, 2008. | MR 2427002 | Zbl 1149.28001

[15] R. L. Moore, Concerning triods in the plane and the junction points of plane continua, Proc. Nat. Acad. Sci. U.S.A. 14 (1928), 85–88. | JFM 54.0630.03 | MR 7481

[16] H. P. Mulholland, On the total variation of a function of two variables, Proc. London Math. Soc. (2) 46 (1940), 290-311. | JFM 66.0235.01 | MR 1830

[17] A. Sard, The measure of the critical values of differentiable maps, Bull. Amer. Math. Soc. 48 (1942), 883–890. | MR 7523 | Zbl 0063.06720

[18] L. Simon, “Lectures on Geometric Measure Theory”, Proceedings of the Centre for Mathematical Analysis, Vol. 3, Australian National University, Centre for Mathematical Analysis, Canberra 1983. | MR 756417 | Zbl 0546.49019

[19] H. Whitney, A function not constant on a connected set of critical points, Duke Math. J. 1 (1935), 514–517. | JFM 61.1117.01 | MR 1545896