Here is the definition of Lipschitz domain given by Wikipedia.

Let n ∈ N, and let Ω be an open subset of Rn. Let ∂Ω denote the boundary of Ω. Then Ω is said to have Lipschitz boundary, and is called a Lipschitz domain, if, for every point p ∈ ∂Ω, there exists a radius r > 0 and a map $h_p$ : $B_r(p)$ → Q such that

(i) $h_p$ is a bijection;

(ii) $h_p$ and $h^{-1}_p $ are both Lipschitz continuous functions;

(iii)$h_p$(∂Ω ∩ Br(p)) = $Q_0$

(iv) $h_p$(Ω ∩ Br(p)) = $Q_+$;

where $B_{r} (p) := \{ x \in \mathbb{R}^{n} | \| x - p \| < r \}$ denotes the n-dimensional open ball of radius r about p,

Q denotes the unit ball B1(0), and

$Q_{0} := \{ (x_{1}, \dots, x_{n}) \in Q | x_{n} = 0 \}$;

$Q_{+} := \{ (x_{1}, \dots, x_{n}) \in Q | x_{n} > 0 \}$.

Then, it says that "a Lipschitz domain (or domain with Lipschitz boundary) is a domain in Euclidean space whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a Lipschitz continuous function."

What I do not understand is the part that says that the boundary of Lipschitz can be thought of as the graph of a Lipschitz continuous function.

What does it mean by the graph of a Lipschitz function? Which Lipschitz function does it talk about? Does it refer to the function $h_p$ as given above?

Please help me understand this!!

  • $\begingroup$ Sometimes, it is a good idea to forget about wikipedia. ;-) See my "answer". $\endgroup$ – Olórin Jan 30 '15 at 1:51

Here's the real and proper definition of a Lipschitz domain. See the local coordinate as a chage of variable in $\mathbb{R}^d$.

A bounded domain $\Omega \subset \mathbb R^d$ with boundary $\Gamma$ is said to be a Lipschitz domain, if there exist constants $\alpha > 0$, $\beta > 0$, and a finite number of local coordinate systems $(x_1^r,x_2^r,\ldots,x_d^r)$, $1 \le r \le R$, and local Lipschitz continuous mappings $$ a_r : \{\hat x^r=(x_2^r,\ldots,x_d^r) \subset \mathbb R^{d-1} \mid |x_i^r|\le \alpha, 2 \le i \le d\} \to \mathbb R $$ such that \begin{align} &\Gamma = \bigcup_{r=1}^R \{(x_1^r,\hat x^r) \mid x_1^r=a_r(\hat x^r), |\hat x^r| <\alpha\}, \\ &\{(x_1^r,\hat x^r) \mid a_r(\hat x^r) < x_1^r < a_r(\hat x^r) + \beta, |\hat x^r| < \alpha\} \subset \Omega, 1 \le r \le R, \\ &\{(x_1^r, \hat x^r) \mid a_r(\hat x^r)-\beta<x_1^r < a_r(\hat x^r), |\hat x^r|<\alpha \} \subset \Omega_{\epsilon}, 1 \le r \le R. \end{align} In particular, the gometrical interpretation of the conditions is that both $\Omega$ and $\Omega_\epsilon$ are locally situated on exactly one side of the boundary $\Gamma$.

  • $\begingroup$ Can you explain a little bit what it means by local coordinate system ? $\endgroup$ – Hua May 4 '17 at 15:38
  • $\begingroup$ It means a way of locating yourself wrt a basis point. Like on a map. $\endgroup$ – Olórin May 4 '17 at 16:32
  • $\begingroup$ Thanks for reply. Can you give a formal definition ? What is the difference between this and the standard Euclidean coordinate $(x_1,x_2,...,x_d)$ ? $\endgroup$ – Hua May 4 '17 at 16:44
  • $\begingroup$ en.wikipedia.org/wiki/Atlas_(topology) $\endgroup$ – Olórin May 4 '17 at 17:14
  • $\begingroup$ Is $\Omega_\epsilon$ the complement of $\Omega$ without the boundary? This seems to me a strange notation. $\endgroup$ – Nathanael Skrepek Apr 5 '19 at 14:09

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