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 .
Such domains are also called strongly Lipschitz domains to contrast them with weakly Lipschitz domains, which are a more general class of domains. A weakly Lipschitz domain is a domain whose boundary is locally flattable by a Lipschitzeomorphism.
Definition [ edit ] Let
n
∈
N
{\displaystyle n\in \mathbb {N} }
Ω
{\displaystyle \Omega }
open subset of
R
n
{\displaystyle \mathbb {R} ^{n}}
∂
Ω
{\displaystyle \partial \Omega }
boundary of
Ω
{\displaystyle \Omega }
Ω
{\displaystyle \Omega }
Lipschitz domain if for every point
p
∈
∂
Ω
{\displaystyle p\in \partial \Omega }
H
{\displaystyle H}
n
−
1
{\displaystyle n-1}
p
{\displaystyle p}
g
:
H
→
R
{\displaystyle g:H\rightarrow \mathbb {R} }
r
>
0
{\displaystyle r>0}
h
>
0
{\displaystyle h>0}
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}}}
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\},}
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\}.}
More generally,
Ω
{\displaystyle \Omega }
weakly Lipschitz if for every point
p
∈
∂
Ω
,
{\displaystyle p\in \partial \Omega ,}
r
>
0
{\displaystyle r>0}
l
p
:
B
r
(
p
)
→
Q
{\displaystyle l_{p}:B_{r}(p)\rightarrow Q}
l
p
{\displaystyle l_{p}}
bijection ;
l
p
{\displaystyle l_{p}}
l
p
−
1
{\displaystyle l_{p}^{-1}}
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}
B
1
(
0
)
{\displaystyle B_{1}(0)}
R
n
{\displaystyle \mathbb {R} ^{n}}
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\}.}
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 ] Dacorogna, B. (2004). Introduction to the Calculus of Variations . Imperial College Press, London. ISBN 1-86094-508-2 . 
