1
$\begingroup$

Suppose $U\subset\mathbb{R}^n$ is open and the boundary $M = \partial U$ is a $C^2$-manifold. Let $\delta$ be the signed distance to $M$.
May I ask under what conditions, without assuming compactness, I can guarantee there exists a uniformly large neighborhood of $M$ for which $\delta\in C^2$.
Regularity of the Distance Function
Distance to $C^k$-hypersurfaces

In particular, if $n(y)$ is the outward unit normal vector, does assuming a bound on $\partial_y n$ suffices? Thank you.

Improve this question
$\endgroup$
3
  • 1
    $\begingroup$ Take $U = \mathbb{R}^3$ and $M= \{z=0\} \cup \{z=\epsilon\}$. You cannot obtain a uniform such estimate as $\epsilon\to 0$ (the distance function is not $C^2$ at $z=\epsilon/2$). $\endgroup$
    – Otis Chodosh
    Commented Mar 15, 2023 at 21:52
  • $\begingroup$ Thank you, but I want to fix $M$. So, given this particular $M$, can I find a uniform $\epsilon>0$ such that the signed distance is $C^2$ for all $y\in \{z:d(z,M) <\epsilon\}$? $\endgroup$
    – Jo'
    Commented Mar 15, 2023 at 22:37
  • 2
    $\begingroup$ you can do something similar where two sheets of M get close together near infinity. Or e.g a spiraling curve. $\endgroup$
    – Otis Chodosh
    Commented Mar 15, 2023 at 23:31

0

Reset to default

You must log in to answer this question.

Browse other questions tagged .