Lipschitz domain

In mathematics, 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. The term is named after the German mathematician Rudolf Lipschitz.

Definition
Let $$n \in \mathbb N$$. Let $$\Omega$$ be a domain of $$\mathbb R^n$$ and let $$\partial\Omega$$ denote the boundary of $$\Omega$$. Then $$\Omega$$ is called a Lipschitz domain if for every point $$p \in \partial\Omega$$ there exists a hyperplane $$H$$ of dimension $$n-1$$ through $$p$$, a Lipschitz-continuous function $$g : H \rightarrow \mathbb R$$ over that hyperplane, and reals $$r > 0$$ and $$h > 0$$ such that where
 * $$\Omega \cap C = \left\{x + y \vec{n} \mid x \in B_r(p) \cap H,\ -h < y < g(x) \right\}$$
 * $$(\partial\Omega) \cap C = \left\{x + y \vec{n} \mid x \in B_r(p) \cap H,\ g(x) = y \right\}$$
 * $$\vec{n}$$ is a unit vector that is normal to $$H,$$
 * $$B_{r} (p) := \{x \in \mathbb{R}^{n} \mid \| x - p \| < r \}$$ is the open ball of radius $$r$$,
 * $$C := \left\{x + y \vec{n} \mid x \in B_r(p) \cap H,\ {-h} < y < h \right\}.$$

In other words, at each point of its boundary, $$\Omega$$ is locally the set of points located above the graph of some Lipschitz function.

Generalization
A more general notion is that of weakly Lipschitz domains, which are domains whose boundary is locally flattable by a bilipschitz mapping. Lipschitz domains in the sense above are sometimes called strongly Lipschitz by contrast with weakly Lipschitz domains.

A domain $$\Omega$$ is weakly Lipschitz if for every point $$p \in \partial\Omega,$$ there exists a radius $$r > 0$$ and a map $$\ell_p : B_r(p) \rightarrow Q$$ such that where $$Q$$ denotes the unit ball $$B_1(0)$$ in $$\mathbb{R}^n$$ and
 * $$\ell_p$$ is a bijection;
 * $$\ell_p$$ and $$l_p^{-1}$$ are both Lipschitz continuous functions;
 * $$\ell_p\left( \partial\Omega \cap B_r(p) \right) = Q_0;$$
 * $$\ell_p\left( \Omega \cap B_r(p) \right) = Q_+;$$


 * $$Q_{0} := \{(x_1, \ldots, x_n) \in Q \mid x_n = 0 \};$$
 * $$Q_{+} := \{(x_1, \ldots, x_n) \in Q \mid x_n > 0 \}.$$

A (strongly) Lipschitz domain is always a weakly Lipschitz domain but the converse is not true. An example of weakly Lipschitz domains that fails to be a strongly Lipschitz domain is given by the two-bricks domain

Applications of Lipschitz domains
Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain. Consequently, many partial differential equations and variational problems are defined on Lipschitz domains.