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

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

where

\vec n

is a unit vector that is normal to

H

,

B_{r}(p):=\{\;x\in \mathbb {R} ^{n}\;|\;\|x-p\|<r\;\}

,

C:=\left\{\;x+y{\vec {n}}\;|\;x\in B_{r}(p)\cap H,\ -h<y<h\;\right\}

.

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},\dots ,x_{n})\in Q\;|\;x_{n}=0\;\};

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

References

Dacorogna, B. (2004). Introduction to the Calculus of Variations. Imperial College Press, London. ISBN 1-86094-508-2.

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