I have a question about a property of Lipschitz domain.

Let $D \subset \mathbb{R}^d$ be a bounded domain (connected open subset ). $D$ is called a bounded Lipschitz domain if there exist positive constants $\delta$, $M$ such that for each $x_0 \in \partial \Omega$ there exist a neighborhood $U_{x_0}$ of $x_0$, local coordinates $y=(y',y_d) \in \mathbb{R}^{d-1} \times \mathbb{R}$, with $y=0$ at $x_0$, and a Lipschitz continous function $f_{x_0}:\mathbb{R}^{d-1} \to \mathbb{R}$, such that \begin{equation*} D \cap U_{x_0}=\{(y',y_N) \in D \cap U_{x_0} \mid y' \in B(0,\delta),\, y_N>f(y') \},\quad \text{Lip}(f) \le M, \end{equation*} where we define $\text{Lip}(f)=\inf \{L \ge 0 \mid |f(x)-f(y)| \le L|x-y|,\, x,y \in B(0,\delta) \}$.

In the above definition, $B(0,\delta)$ denotes the open ball in $\mathbb{R}^d$ centered at the origin with radius $1$.

My question

We denote by $m$ the Lebesgue measure on $D$.

Let $D \subset \mathbb{R}^d$ be a bounded Lipschitz domain. Then, can we show the following?

There exists a positive constant $C \ge 1$ such that $$C^{-1}r^d \le m(\bar{D} \cap B(x,r)) \le Cr^d$$ for any $x \in \bar{D}$, $r \in (0,\text{diam}(D)]$.

It is clear that $m(\bar{D} \cap B(x,r)) \le Cr^d$ for any $x \in \bar{D}$, $r \in (0,\text{diam}(D)]$.

If you know a proof or references, please let me know.

  • $\begingroup$ You can show (more or less directly from your boxed definition) that $D$ satisfies a uniform interior cone condition, which implies your lower bound in a straightforward way. $\endgroup$
    – user101142
    Commented Aug 19, 2018 at 17:44
  • $\begingroup$ I'm sorry but I do not know the uniform interior cone condition well. Can you give me detailed proof? $\endgroup$
    – sharpe
    Commented Aug 19, 2018 at 18:00

1 Answer 1


This is probably somewhat over-the-top, but anyway: The nice paper [1] by Hajlasz, Koskela and Tuominen says that your desired inequality is true for Sobolev extension domains, so for domains $D$ for which there exists a continuous linear operator $E \colon W^{1,p}(D) \to W^{1,p}(\mathbb{R}^n)$ such that $(Eu)_{\restriction D} = u$. It is a classical result that a Lipschitz domain is such an extension domain, e.g. Theorem in Grisvard [2].

[1] Hajłasz, Piotr; Koskela, Pekka; Tuominen, Heli, Sobolev embeddings, extensions and measure density condition, J. Funct. Anal. 254, No. 5, 1217-1234 (2008). ZBL1136.46029.

[2] Grisvard, Pierre, Elliptic problems in nonsmooth domains, Classics in Applied Mathematics 69. Philadelphia, PA: Society for Industrial and Applied Mathematics (SIAM) (ISBN 978-1-611972-02-3/pbk). xx, 410 p. (2011). ZBL1231.35002.

  • $\begingroup$ Thank you for your comment. That became an extremely useful resource for me. $\endgroup$
    – sharpe
    Commented Aug 19, 2018 at 19:08

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge you have read our privacy policy.

Not the answer you're looking for? Browse other questions tagged or ask your own question.