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 ${\displaystyle n\in \mathbb {N} }$. Let ${\displaystyle \Omega }$ be an open subset of ${\displaystyle \mathbb {R} ^{n}}$ and let ${\displaystyle \partial \Omega }$ denote the boundary of ${\displaystyle \Omega }$. Then ${\displaystyle \Omega }$ is called a Lipschitz domain if for every point ${\displaystyle p\in \partial \Omega }$ there exists a hyperplane ${\displaystyle H}$ of dimension ${\displaystyle n-1}$ through ${\displaystyle p}$, a Lipschitz-continuous function ${\displaystyle g:H\rightarrow \mathbb {R} }$ over that hyperplane, and the values ${\displaystyle r>0}$ and ${\displaystyle h>0}$ such that

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

where

${\displaystyle {\vec {n}}}$ is a unit vector that is normal to ${\displaystyle H,}$
${\displaystyle B_{r}(p):=\{x\in \mathbb {R} ^{n}\mid \|x-p\|
${\displaystyle C:=\left\{x+y{\vec {n}}\mid x\in B_{r}(p)\cap H,\ -h

More generally, ${\displaystyle \Omega }$ is said to be weakly Lipschitz if for every point ${\displaystyle p\in \partial \Omega ,}$ there exists a radius ${\displaystyle r>0}$ and a map ${\displaystyle l_{p}:B_{r}(p)\rightarrow Q}$ such that

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

where ${\displaystyle Q}$ denotes the unit ball ${\displaystyle B_{1}(0)}$ in ${\displaystyle \mathbb {R} ^{n}}$ and

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

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.

References

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