Boundary of the set of critical points

Question

Let $u\in C^\infty(\mathbb{R}^d)$ and consider $$ E = \partial\{x:Du(x)=0\}. $$ I am interested in understanding whether $\mu(E)=0$ or not, $\mu$ being the $d$-dimensional Lebesgue measure. What I could prove so far is that there exists a point $x\in E$ such that, for all neighborhoods $U\in U(x)$ we have $$ \mu(E\cap \overline{U})>0. $$ Any idea on how to proceed with proving that $\mu(E)>0$, assumed that it is possible?

Answer 1

The measure $\mu(E)$ may be positive or zero. It is easy to give examples of the case $\mu(E) = 0$, so below is an example in $C^\infty(\mathbb{R})$ with $\mu(E) > 0$.

In light of this post here, it suffices to exhibit a closed subset of $\mathbb{R}^n$ whose boundary has positive measure. In the spirit of what you pointed out in your question, the so-called Fat Cantor Set is closed with empty interior (hence it is its own boundary). Moreover, by computing a simple geometric sum one sees that it has positive measure.