0
$\begingroup$

I use a lot in the study of pde bounded Lipschitz domains $\Omega\subseteq\mathbb{R}^N$. However I have noticed that there are some major differences in their definitions. I will put here two of them, given their references too:

1. The first definition which is the one that I prefer is taken from Antoine Henrot, Michel Pierre - Shape variation and optimization, page 56, def. 2.4.5

enter image description here

2. The second definition which seems less restrictive is taken from Geovanni Leoni - A first course in Sobolev spaces, page 354, def. 12.9 and def.12.10

The boundary $\partial \Omega$ of an open set $\Omega \subset \mathbb{R}^N$ is $locally $ $Lipschitz$ if for each $x_0 \in \partial \Omega$ there exist a neighborhood $A$ of $x_0$, local coordinates $y=(y',y_N) \in \mathbb{R}^{N-1} \times \mathbb{R}$, with $y=0$ at $x=x_0$, a Lipschitz function $f_x:\mathbb{R}^{N-1} \to \mathbb{R}$ of Lipschitz constant $L_x>0$, and $r_x>0$, such that \begin{equation*} \Omega \cap A=\{(y',y_N) \in \Omega \cap A: y' \in Q(0,r), y_N>f(y') \}, \end{equation*} where $Q(0,r_x)$ is the $N-1$ dimensional open ball centered at the origin with radius $r_x>0$.

Note that:

(i) It can be shown easily that if $\Omega$ is bounded then (by the fact that $\partial\Omega$ is a compact set) the constants $r_x$ and $L_x$ can be chosen equal for any $x\in\partial\Omega$. Practically any bounded locally Lipschitz domain is a uniform Lipschitz domain (see also Exercise 12.11 in Leoni's book cited above). For a nice proof of Geovanni Leoni given on math.stackexchange see here: Locally Lipschitz, boundary

(ii) The first definition says what is a uniform Lipschitz domain. A local version will be the same but with the constants $L=L_x,r=r_x$ and $a=a_x$ all depending on $x$. This local version is used for example in this course available online:

  • PASCAL FREY, YANNICK PRIVAT - Introduction a l'optimisation de forme et application a la mecanique de fluides, page 2. It can be found here.

Here are my questions?

1) Is it true that any bounded (locally) Lipschitz domain (in the sense of the first definition) is a uniform Lipschitz domain (in the sense of the first definition too)?

2) Are the two definition equivalent (the first one seen with $L,a,r$ depending on $x$) in the case of bounded Lipschitz domains?

What I have done?

  1. I could find $L$ and $r$ by the same idea as in the proof of Leoni posted on MathStackExchange but all my trials failed when it comed to set $a$. It is very hard to control that $a$.

  2. It's clear that the first imply the second. The other implication is harder, but what is easy to prove is that the second equality on the first definition can be obtain from the equality given on the second definition.

However I beleive that this is true even though I didn't figured it out how. Any advice will be great!

Improve this question
$\endgroup$
12
  • $\begingroup$ Regarding Q2: If $L$ is uniform, then is $a$ not automatically uniform, too? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jan 30, 2022 at 14:39
  • $\begingroup$ It may depend on the system of coordinates that we choose, no? I mean that a function of Lipschitz constant $L$ in some system of coordinates can be a Lipschitz function of a bigger constant in other system of coordinates. $\endgroup$
    – Bogdan
    Commented Jan 30, 2022 at 14:43
  • 1
    $\begingroup$ It may even fail to be Lipschitz, by why does that matter? If you already know that at every point the boundary is a graph of an $L$-Lipschitz function within the $r$-neighbourhood of that point, then just set $a = L r$. Or am I missing something? $\endgroup$
    – Mateusz Kwaśnicki
    Commented Jan 30, 2022 at 15:02
  • $\begingroup$ We set $L$ from a finite cover of $\partial\Omega$. I defined $r$ in terms of $a_{x_i}$. If some of the $a_{x_i}$ are smaller than $Lr$ what should we do? If we change $r$ we change $a=Lr$ and vice-versa. $\endgroup$
    – Bogdan
    Commented Jan 30, 2022 at 15:30
  • $\begingroup$ I defined $r=\min(\rho_{x_i}),\ i\in\{1,...n\}$ (from the finite cover) where $\rho_{x}=\min\{r_x/2,\sup\{\rho>0\ |\ B_{N-1}(x,\rho)\subset\varphi_x^{-1}(-a_x,a_x)\}\}$ for each $x\in\partial\Omega$. $\endgroup$
    – Bogdan
    Commented Jan 30, 2022 at 15:33

1 Answer 1

Reset to default
1
$\begingroup$

The second condition does not imply the first condition. Have a look at Figure 1.3 in Grisvard's Elliptic Problems in Nonsmooth Domains. Grisvard takes a triangle, then "crumples" it so that there is no ray from one of the corners which remains in the triangle.

Improve this answer
$\endgroup$