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 ${\displaystyle n\in \mathbb {N} }$. Let ${\displaystyle \Omega }$ be a domain 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 reals ${\displaystyle r>0}$ and ${\displaystyle h>0}$ such that

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

where

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

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

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

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

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

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

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

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

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 ^[1]

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

1. Werner Licht, M. "Smoothed Projections over Weakly Lipschitz Domains", arXiv, 2016.

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