On the Morse-Sard Property and Level Sets of Sobolev and BV Functions

TL;DR

This paper proves Luzin N and Morse-Sard properties for BV and Sobolev functions in the plane, showing that most level sets are unions of Lipschitz or C^1 arcs with controlled tangent vectors.

Contribution

It establishes new regularity and measure-theoretic properties for level sets of BV and W^{2,1} functions, extending classical results to these function spaces.

Findings

Almost all level sets of BV_2-functions are unions of Lipschitz arcs.

For W^{2,1}-functions, level sets are unions of C^1 arcs with absolutely continuous tangent vectors.

The results generalize Morse-Sard theorem to BV and Sobolev functions in the plane.

Abstract

We establish Luzin $N$ and Morse-Sard properties for $BV_2$-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 $W^{2,1}$-functions we strengthen the conclusion and show that almost all level sets are finite disjoint unions of $C^1$-arcs whose tangent vectors are absolutely continuous.