JournalsrmiVol. 29, No. 1pp. 1–23

On the Morse–Sard property and level sets of Sobolev and BV functions

  • Jean Bourgain

    Institute for Advanced Study, Princeton, United States
  • Jan Kristensen

    Oxford University, United Kingdom
  • Mikhail V. Korobkov

    Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation
On the Morse–Sard property and level sets  of Sobolev and BV functions cover
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Abstract

We establish Luzin NN and Morse–Sard properties for BV2\mathrm{BV}_2 functions defined on open domains in the plane. Using these results we prove that almost all level sets are finite disjoint unions of Lipschitz arcs whose tangent vectors are of bounded variation. In the case of W2,1\mathrm{W}^{2,1} functions we strengthen the conclusion and show that almost all level sets are finite disjoint unions of C1\mathrm{C}^1 arcs whose tangent vectors are absolutely continuous along these arcs.

Cite this article

Jean Bourgain, Jan Kristensen, Mikhail V. Korobkov, On the Morse–Sard property and level sets of Sobolev and BV functions. Rev. Mat. Iberoam. 29 (2013), no. 1, pp. 1–23

DOI 10.4171/RMI/710