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[edit]

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

* Ω ∩ C = { x + y n -> ∣ x ∈ B r ( p ) ∩ H , − h < y < g ( x ) } {\displaystyle \Omega \cap C=\left\\{x+y{\vec {n}}\mid x\in B_{r}(p)\cap H,\ -h<y<g(x)\right\\}}
* ( ∂ Ω ) ∩ C = { x + y n -> ∣ x ∈ B r ( p ) ∩ H , g ( x ) = y } {\displaystyle (\partial \Omega )\cap C=\left\\{x+y{\vec {n}}\mid x\in B_{r}(p)\cap H,\ g(x)=y\right\\}}

where

n -> {\displaystyle {\vec {n}}} is a unit vector that is normal to  H , {\displaystyle H,}
B r ( p ) := { x ∈ R n ∣ ‖ x − p ‖ < r } {\displaystyle B_{r}(p):=\\{x\in \mathbb {R} ^{n}\mid \|x-p\|<r\\}} is the open ball of radius  r {\displaystyle r} ,
C := { x + y n -> ∣ x ∈ B r ( p ) ∩ H , − h < y < h } . {\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,  Ω {\displaystyle \Omega } is locally the set of points located above the graph of some Lipschitz function.

## Generalization[edit]

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  Ω {\displaystyle \Omega } is weakly Lipschitz if for every point  p ∈ ∂ Ω , {\displaystyle p\in \partial \Omega ,} there exists a radius  r > 0 {\displaystyle r>0} and a map  l p : B r ( p ) -> Q {\displaystyle l_{p}:B_{r}(p)\rightarrow Q} such that

* l p {\displaystyle l_{p}} is a bijection;
* l p {\displaystyle l_{p}} and  l p − 1 {\displaystyle l_{p}^{-1}} are both Lipschitz continuous functions;
* l p ( ∂ Ω ∩ B r ( p ) ) = Q 0 ; {\displaystyle l_{p}\left(\partial \Omega \cap B_{r}(p)\right)=Q_{0};}
* l p ( Ω ∩ B r ( p ) ) = Q + ; {\displaystyle l_{p}\left(\Omega \cap B_{r}(p)\right)=Q_{+};}

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

Q 0 := { ( x 1 , … , x n ) ∈ Q ∣ x n = 0 } ; {\displaystyle Q_{0}:=\\{(x_{1},\ldots ,x_{n})\in Q\mid x_{n}=0\\};}
Q + := { ( x 1 , … , x n ) ∈ Q ∣ 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[edit]

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[edit]

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.