  For a regular (in a sense) mapping $v:\mathbb{R}^n \to \mathbb{R}^d$ we study
the following problem: {\sl let $S$ be a subset of $m$-critical a set $\tilde
Z_{v,m}=\{{\rm rank} \nabla v\le m\}$ and the equality $\mathcal{H}^\tau(S)=0$
(or the inequality $\mathcal{H}^\tau(S)<\infty$) holds for some $\tau>0$. Does
it imply that $\mathcal{H}^{\sigma}(v(S))=0$ for some $\sigma=\sigma(\tau,m)$?}
(Here $\mathcal{H}^\tau$ means the $\tau$-dimensional Hausdorff measure.) For
the classical classes $C^k$-smooth and $C^{k+\alpha}$-Holder mappings this
problem was solved in the papers by Bates and Moreira. We solve the problem for
Sobolev $W^k_p$ and fractional Sobolev $W^{k+\alpha}_p$ classes as well. Note
that we study the Sobolev case under minimal integrability assumptions
$p=\max(1,n/k)$, i.e., it guarantees in general only {\it the continuity} (not
everywhere differentiability) of a mapping. In particular, there is an
interesting and unexpected analytical phenomena here: if $\tau=n$ (i.e., in the
case of Morse--Sard theorem), then the value $\sigma(\tau)$ is the same for the
Sobolev $W^k_p$ and for the classical $C^k$-smooth case. But if $\tau<n$, then
the value $\sigma$ depends on $p$ also; the value $\sigma$ for $C^k$ case could
be obtained as the limit when $p\to\infty$. The similar phenomena holds for
Holder continuous $C^{k+\alpha}$ and for the fractional Sobolev
$W^{k+\alpha}_p$ classes. The proofs of the most results are based on our
previous joint papers with J. Bourgain and J. Kristensen (2013, 2015). We also
crucially use very deep Y. Yomdin's entropy estimates of near critical values
for polynomials (based on algebraic geometry tools).

Analysis of PDEs