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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 $\lvert r D^2 r \rvert < 1$.

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.

In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made on page 60.

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 guaranteed to be smooth.

How is the inequality $(1)$ proved to hold on the entirety of $U$? Going to the paper of Evans and Spruck, as Ilmanen suggests, I could only make it work in a thin neighborhood of $(Q_t)$, say small enough that the spatial Hessian has $r \lvert D^2 r \rvert < 1$.

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 $\lvert r D^2 r \rvert < 1$.

In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made on page 60.

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 guaranteed to be smooth.

How is this inequality proved to hold on the entirety of $U$? I could only prove it on a thin tubular neighborhood surrounding $(Q_t)$—see below.

He sends the reader to a paper of Evans and Spruck, where it is proved that $r$ satisfies the PDE \begin{equation} \frac{\partial r}{\partial t} - \sum_{i=1}^n \frac{\lambda_i}{1 - \lambda_i r} = 0, \end{equation} where $\lambda_i = \lambda_i(x,t)$, $i = 1,\dots,n$ are the eigenvalues of the Hessian $D^2 r(\cdot,t)$ at $x$. From this I get that \begin{equation} r\Big(\frac{\partial r}{\partial t} - \Delta r\Big) = \sum_{i=1}^n \frac{(\lambda_i r)^2}{1 - \lambda_i r}, \end{equation} which is indeed non-negative provided $\lambda_i r < 1$.

This can be ensured by working in a thin tubular neighborhood of $(Q_t)$, but in principle this could fail if there were a 'large' $U$ so that $r$ is smooth on $U$, but $\lambda_i r \not < 1$ in $U$.

In Ilmanen's monograph on elliptic regularization and mean curvature flow, the following claim is made on page 60.

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 guaranteed to be smooth.

How is the inequality $(1)$ proved to hold on the entirety of $U$? Going to the paper of Evans and Spruck, as Ilmanen suggests, I could only make it work in a thin neighborhood of $(Q_t)$, say small enough that the spatial Hessian has $r \lvert D^2 r \rvert < 1$.