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 Germany mathematician Rudolf Lipschitz.

Definition

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

• $\mathrm{\Omega }\cap C=\{x+y\overrightarrow{n}\mid x\in B_r(p)\cap H,-h<y<g(x)\}$
• $(\partial \mathrm{\Omega })\cap C=\{x+y\overrightarrow{n}\mid x\in B_r(p)\cap H,g(x)=y\}$

where

$\overrightarrow{n}$ is a unit vector that is normal to $H,$

$B_r(p):=\{x\in {\mathbb{ℝ}}^n\mid \Vert x-p\Vert <r\}$ is the open ball of radius $r$,

$C:=\{x+y\overrightarrow{n}\mid x\in B_r(p)\cap H,-h<y<h\}.$

In other words, at each point of its boundary, $\mathrm{\Omega }$ 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 $\mathrm{\Omega }$ is weakly Lipschitz if for every point $p\in \partial \mathrm{\Omega },$ there exists a radius $r>0$ and a map $l_p:B_r(p)\to Q$ such that

• $l_p$ is a bijection;
• $l_p$ and $l_p^{-1}$ are both Lipschitz continuous functions;
• $l_p(\partial \mathrm{\Omega }\cap B_r(p))=Q_0;$
• $l_p(\mathrm{\Omega }\cap B_r(p))=Q_{+};$

where $Q$ denotes the unit ball $B_1(0)$ in ${\mathbb{ℝ}}^n$ and

$Q_0:=\{(x_1,\dots ,x_n)\in Q\mid x_n=0\};$

$Q_{+}:=\{(x_1,\dots ,x_n)\in Q\mid x_n>0\}.$

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.