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In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made. (Just for reference, this is on page 60.) For the proof—apparently standard calculations—the reader is referred to a paper of Evans and Spruck.

Claim. If $(Q_t \mid t_0 \leq t \leq t_1)$ is a smooth mean curvature flow in $\mathbf{R}^n$, then the signed distance function $r$ to $Q_t$ satisfies the inequality \begin{equation} \tag{1} r\Big(\frac{\partial r}{\partial t} - \Delta r \Big) \geq 0 \end{equation} on the open neighborhood $U$ of $(Q_t)$ where $r$ is smooth.

Does the inequality $(1)$ really hold on the entirety of $U$?

Going through the Evans–Spruck paper, I could only make it work in a thin neighborhood of $(Q_t)$, say small enough that the spatial Hessian has $\lvert r D^2 r \rvert < 1$.

For the argument that follows in Ilmanen's book, it's perfectly fine to work with a thinner neighborhood of $(Q_t)$. What my question is getting at is whether it's necessary to update $U$ to a smaller open set in order for $(1)$ to hold.

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This computation is only used when computing the evolution of the function $\phi$ defined on the previous page and $\phi$ is defined so it is supported in a small neighborhood of the smooth solution.

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  • $\begingroup$ Sure, but the size of the neighborhood isn't updated - the only thing imposed on it is that it be narrow enough that the Brakke flow and $Q$ do not initially intersect there, and that $r$ be smooth on it. $\endgroup$
    – Leo Moos
    Dec 15, 2022 at 8:10
  • $\begingroup$ Why would it need to be updated? You are trying to get avoidance on a compact time interval and have a smooth compact flow so you just choose $\gamma$ small enough initially so the differential inequality is satisfied (I guess this should have been made more explicit and may be a gap/oversight in the proof but seems easy to fix). $\endgroup$
    – RBega2
    Dec 15, 2022 at 15:28
  • $\begingroup$ I meant update in the sense that one might have to change $\gamma$ that was chosen on the previous page, 'so small that [...] $r$ is smooth on the set $U$', to a smaller value. I didn't mean that $\gamma$ would have to be updated along the flow. (Is that how you read my comment? That's how I interpreted your reply, anyway.) I didn't mean the question as a gotcha - obviously the size of $U$ plays no role in the argument, which is fine as is. I was just wondering whether I'd missed something, and the inequality really does hold wherever $r$ is smooth. $\endgroup$
    – Leo Moos
    Dec 15, 2022 at 15:58

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