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