Revista Matemática Iberoamericana


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Volume 29, Issue 1, 2013, pp. 1–23
DOI: 10.4171/RMI/710

Published online: 2013-01-14

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

Jean Bourgain[1], Mikhail V. Korobkov[2] and Jan Kristensen[3]

(1) Institute for Advanced Study, Princeton, United States
(2) Siberian Branch of the Russian Academy of Sciences, Novosibirsk, Russian Federation
(3) Oxford University, United Kingdom

We establish Luzin $N$ and Morse–Sard properties for $\mathrm{BV}_2$ 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 $\mathrm{W}^{2,1}$ functions we strengthen the conclusion and show that almost all level sets are finite disjoint unions of $\mathrm{C}^1$ arcs whose tangent vectors are absolutely continuous along these arcs.

Keywords: BV2 and W2,1 functions, Luzin N property, Morse–Sard property, level sets

Bourgain Jean, Korobkov Mikhail, Kristensen Jan: On the Morse–Sard property and level sets of Sobolev and BV functions. Rev. Mat. Iberoamericana 29 (2013), 1-23. doi: 10.4171/RMI/710