# 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.

Such domains are also called strongly Lipschitz domains to contrast them with weakly Lipschitz domains, which are a more general class of domains. A weakly Lipschitz domain is a domain whose boundary is locally flattable by a Lipschitzeomorphism.

## Definition

Let $n\in \mathbb {N}$ . Let $\Omega$ be an open subset 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 the values $r>0$ and $h>0$ such that

• $\Omega \cap C=\left\{x+y{\vec {n}}\mid x\in B_{r}(p)\cap H,\ -h • $(\partial \Omega )\cap C=\left\{x+y{\vec {n}}\mid x\in B_{r}(p)\cap H,\ g(x)=y\right\}$ where

${\vec {n}}$ is a unit vector that is normal to $H,$ $B_{r}(p):=\{x\in \mathbb {R} ^{n}\mid \|x-p\| $C:=\left\{x+y{\vec {n}}\mid x\in B_{r}(p)\cap H,\ -h More generally, $\Omega$ is said to be weakly Lipschitz if for every point $p\in \partial \Omega ,$ there exists a radius $r>0$ and a map $l_{p}:B_{r}(p)\rightarrow Q$ such that

• $l_{p}$ is a bijection;
• $l_{p}$ and $l_{p}^{-1}$ are both Lipschitz continuous functions;
• $l_{p}\left(\partial \Omega \cap B_{r}(p)\right)=Q_{0};$ • $l_{p}\left(\Omega \cap B_{r}(p)\right)=Q_{+};$ where $Q$ denotes the unit ball $B_{1}(0)$ in $\mathbb {R} ^{n}$ and

$Q_{0}:=\{(x_{1},\ldots ,x_{n})\in Q\mid x_{n}=0\};$ $Q_{+}:=\{(x_{1},\ldots ,x_{n})\in Q\mid x_{n}>0\}.$ ## 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.