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Lipschitz domain

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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 ∈ 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

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 \ell _{p}:B_{r}(p)\rightarrow Q} such that

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

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.
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