= Gradient of a Sobolev function vanishes on a level set
{title2=$Du=0\text{ a.e. on }\{u=t\}$}
If $u\in W^{1,2}_{\mathrm{loc}}$, then $Du=0$ <almost everywhere> on every <level set> $\{u=t\}$. Choose smooth truncations $\Phi_\varepsilon(s)$ with $|\Phi_\varepsilon|\leq2\varepsilon$, derivative one near zero, bounded derivative, and derivative zero outside $(-2\varepsilon,2\varepsilon)$. The <Sobolev chain rule> gives $D\Phi_\varepsilon(u-t)\to\mathbf1_{\{u=t\}}Du$ in local $L^2$ by <dominated convergence theorem>, while $\Phi_\varepsilon(u-t)\to0$. The limiting <weak derivative> is therefore zero.
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