Lipschitz domain

Remarks

It follows that $\partial\Omega \cap B(p, r) = \set{x \in B(p, r) : x_n = \varphi(x')}$. Many authors also require $\Omega$ to be bounded, and sometimes connected. If $\Omega$ is bounded, finitely many such charts cover $\partial\Omega$, so one can use a common radius and a common Lipschitz constant. These numbers are called the Lipschitz character of $\Omega$. Example: the square $(0, 1)^2$. Near an edge point the square is the region above a horizontal line. Near the corner $(0, 0)$, in coordinates rotated by $45$ degrees, it is the region $\set{x_2 > \abs{x_1}}$ near the corner, and $\abs{x_1}$ is Lipschitz with constant $1$. Non-example: the slit disc $\set{\abs{z} < 1} \setminus [0, 1)$ in $\R^2$. Near the point $1/2$ of the slit, the set lies on both sides of the slit, so it cannot be the region on one side of a graph. Every bounded open set with $C^1$ boundary is a Lipschitz domain.