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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 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.

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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.

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  • $\begingroup$ thank you Gerw, I will check that out. I was hoping to avoid this kind of things but I convinced myself that there is no easy way out... $\endgroup$ – leo monsaingeon Nov 14 '13 at 9:47

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