Lipschitz domain

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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.

[edit] Definition

Let $n \in \mathbb{N}$ , and let $Ω$ be an open and bounded subset of $\mathbb{R}^{n}$ . Let $\partial \Omega$ denote the boundary of $Ω$ . Then $Ω$ is said to have Lipschitz boundary, and is called a Lipschitz domain, if, for every point $p \in \partial \Omega$ , there exists a radius $r > 0$ and a map $h_{p} : B_{r} (p) \to Q$ such that

$h p$ is a bijection;
$h p$ and $h_{p}^{-1}$ are both Lipschitz continuous functions;
$h_{p}^{-1} \left( \partial \Omega \cap B_{r} (p) \right) = Q_{0}$ ;
$h_{p}^{-1} \left( \Omega \cap B_{r} (p) \right) = Q_{+}$ ;

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 $B 1 (0)$ , and

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

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

[edit] 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.