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Suppose I have a sequence $u_n \to u$ in $H^1_0(\Omega)$ on a smooth and bounded domain. For some $p>1$ and $s \in (0,\frac 12)$, is it possible to estimate the norm of the characteristic function of the zero level set of $u_n$, $$\lVert \chi_{\{u_n=0\}}\rVert_{W^{s,p}(\Omega)}$$ in terms of norms of $u_n$? In particular I am looking for a uniform bound for the above expression.

We know it belongs eg to $W^{\epsilon, 2}(\Omega)$ for $\epsilon < \frac 12$, I just want to know if it can be bounded uniformly.

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  • $\begingroup$ For the set $\{u_n=0\}$ to make sense, you need something, say continuity of $u_n$, which does not follow from your assumption of regularity when the dimension is $\ge 2$. $\endgroup$
    – Bazin
    Commented Mar 14, 2020 at 17:03
  • $\begingroup$ @Bazin let us assume $n \leq 3$ and $u_n, u$ belong to $H^2(\Omega)$, so they are all continuous functions $\endgroup$
    – StopUsingFacebook
    Commented Mar 14, 2020 at 18:36

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