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

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

where

${\vec {n}}$ 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

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

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

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

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

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.

References

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