# Regularity of a shrunken domain

I am encountering a geometrical question that intuitively seems obvious but I have a lack of argument to prove or disprove it in a rigorous manner.

Let $$\Omega\subset\Bbb R^d$$ be an open bounded (may be connected just to make it simpler)

For $$\delta>0$$ small enough we define the shrunken version of $$\Omega$$ as by $$\Omega_\delta= \{x\in \Omega~: \operatorname{dist}(x,\partial \Omega)>\delta\}$$

Basically, for $$\delta$$ small enough, $$\Omega$$ and $$\Omega_\delta$$ have a similar shape. For instance if $$\Omega= B(0,1)$$ is the unit ball then, $$\Omega_\delta=B(0,\delta) =\delta B(0,1)$$ is the ball centered at 0 with radius $$\delta$$.

Definition An open set $$\Omega$$ is said to satisfy the Volum Density Condition(VDC) if there exists a constant $$\kappa>0$$ such that for all $$x\in \partial \Omega$$ and $$r\in (0,1)$$ $$|\Omega\cap B(x,r)|\geq \kappa r^d.$$

An open set $$\Omega$$ is said to be of class $$C^k$$ if for every $$x\in \partial \Omega$$ there exists $$r>0$$ and a mapping $$\gamma: \Bbb R^{d-1}\to\Bbb R$$ such that

$$\Omega\cap B(x,r)= \{x=(x',x_d)\in B(x,r)~: x_d>\gamma(x')\}$$

A natural question would be: Does $$\Omega_\delta$$ inherit the regularity properties of $$\Omega$$? Precisely,

1)If $$\Omega$$ is $$C^k$$ do we have that $$\Omega_\delta$$ is also $$C^k$$?

2) If $$\Omega$$ satisfies the Volum Density Condition(VDC) does $$\Omega_\delta$$ also satisfies the Volum Density Condition(VDC)?

• Question 1 has a clear answer: yes if $k \geqslant 2$ and $\delta > 0$ is small enough. This is given in most textbooks on PDE's, I suppose, I can search for an exact reference if you like. However, if $\delta > 0$ is too large, then smoothness of $\Omega_\delta$ may deteriorate: consider, say, a unit ball and $\delta = 1$. Also, if $k = 1$, then $\Omega_\delta$ may become highly irregular even when $\delta > 0$ is small. An example: $\Omega$ a region above the graph of $y = |x| / \log(1/|x|)$ near $x = 0$. – Mateusz Kwaśnicki Jun 7 at 21:02
• @MateuszKwaśnicki Please I would like to see does reference. Do you have an idea about what could happen with VDC condition? I technically like that condition very much. – Guy Fsone Jun 7 at 21:06
• Of course, large $\delta$ are not interesting at all. – Guy Fsone Jun 7 at 21:08
• In the example of Mateusz Kwasnicki, the "highly irregular" means "having an outward cusp", and such things do not satisfy VDC (while $C^1$ domains to, of course). – Kostya_I Jun 7 at 21:24
• @GuyFsone: Section 14.6 in Gilbarg and Trudinger has what you need. – Mateusz Kwaśnicki Jun 7 at 21:35