= Sobolev gradient vanishes on a zero set
{c}
{title2=$\nabla w=0\text{ a.e. on }\{w=0\}$}
For a <Sobolev space> function, almost every line parallel to a coordinate axis gives an absolutely continuous restriction. An absolutely continuous function has <derivative> zero <almost everywhere> on each of its level sets. Slicing and <Fubini's theorem> give the assertion in higher dimension. If $w\in W^{2,p}$, apply it to $w$ and then to each component of its <gradient> to get $D^2w=0$ <almost everywhere> on $\{w=0\}$. This supplies the contact differentiation used in <noncontact of ROF level boundaries>.
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