I'm wondering if there are known results about the "regularity" (in some sense to be determined) of sub and super levelsets of Sobolev functions $u\in W^{1,p}(\mathbb{R}^d)$. More precisely:


Assume $u\in W^{1,p}(\mathbb{R}^d)$, fix a constant $M\in \mathbb{R}$, and let
$$
E\_M:=\{x\in \mathbb{R}^d,\hspace{1cm}u(x)>M\}\subset \mathbb{R}^d
$$
be the corresponding super-levelset of $u$. What can we say about the regularity of $E\_M$? I actually have no precise idea of what "regularity" should mean here, but I am not talking about regularity of the boundary $\partial E\_M$ as in the usual context of PDE's...


I think a natural question to ask is for example: how can one guarantee that $E\_M$ is open (for a suitable representative of $u$), or at least that $E\_M$ has nonempty interior? For example if $p$ is large enough we know by Sobolev inequality that $u$ is continuous so $E\_M$ is open. When the gradient regularity deteriorates, how "ugly" could $E\_M$ be? While still being measurable $E\_M$ could be quite pathological, for example a [Smith-Volterra-Cantor set](http://en.wikipedia.org/wiki/Smith%E2%80%93Volterra%E2%80%93Cantor_set) with positive measure but empty interior (well, maybe not, this is actually my question!)


Also, what if we replace $u\in W^{1,p}$ by a weaker difference-quotient-type condition
$$
u\in L^p\quad\text{and}\quad \sup\limits\_{0\neq h\in R^d}\frac{|\tau\_h u-u|\_{L^p(\mathbb{R}^d)}}{|h|^{\alpha}}<\infty \text{ for some }\alpha\in(0,1),
$$
where $\tau\_h u(.)=u(.-h)$ denotes the usual shift???


Thank you in advance for your reference suggestions, input, and comments.



sub and super-levelset regularity for Sobolev functions
One positive answer is that this set is $p$-quasi-open, see some resource about capacity theory, e.g., here: <https://math.stackexchange.com/questions/48776/capacity-theory-beginner-resources>.


