On the Morse-Sard property and level sets of $W^{n,1}$ Sobolev functions on ${\mathbb R}^n$

TL;DR

This paper proves Luzin N and Morse--Sard properties for Sobolev functions in $W^{n,1}$, showing that almost all level sets are finite unions of smooth manifolds, with extensions to BV functions.

Contribution

It establishes new Luzin--type approximation results for Sobolev and BV functions with small Hausdorff content exceptional sets.

Findings

Almost all level sets are finite unions of $C^1$-smooth manifolds.

Luzin N and Morse--Sard properties hold for $W^{n,1}$ functions.

Results extend to functions of bounded variation $BV_n$.

Abstract

We establish Luzin N and Morse--Sard properties for functions from the Sobolev space $W^{n,1}({\mathbb R}^{n})$. Using these results we prove that almost all level sets are finite disjoint unions of $C^1$--smooth compact manifolds of dimension $n-1$. These results remain valid also within the larger space of functions of bounded variation $BV_{n}({\mathbb R}^{n})$. For the proofs we establish and use some new results on Luzin--type approximation of Sobolev and BV--functions by $C^k$--functions, where the exceptional sets have small Hausdorff content.