On the Morse–Sard property and level sets of W n ,1 Sobolev functions on ℝ n

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