Lipschitz Boundary

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 $n \in \mathbb N$. Let $\Omega$ be a domain of $\mathbb R^n$ and let $\partial\Omega$ denote the boundary of $\Omega$. Then $\Omega$ is called a Lipschitz domain if for every point $p \in \partial\Omega$ there exists a hyperplane $H$ of dimension $n-1$ through $p$, a Lipschitz-continuous function $g : H \rightarrow \mathbb R$ over that hyperplane, and reals $r > 0$ and $h > 0$ such that
* $\Omega \cap C = \left\$
* $(\partial\Omega) \cap C = \left\$
where
:$\vec$ is a unit vector that is normal to $H,$
:$B_ (p) := \$ is the open ball of radius $r$,
:$C := \left\.$

In other words, at each point of its boundary, $\Omega$ 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 $\Omega$ is weakly Lipschitz if for every point $p \in \partial\Omega,$ there exists a radius $r > 0$ and a map $l_p : B_r(p) \rightarrow Q$ such that
* $l_p$ is a bijection;
* $l_p$ and $l_p^$ are both Lipschitz continuous functions;
* $l_p\left( \partial\Omega \cap B_r(p) \right) = Q_0;$
* $l_p\left( \Omega \cap B_r(p) \right) = Q_+;$
where $Q$ denotes the unit ball $B_1(0)$ in $\mathbb^$ and

:$Q_ := \;$
:$Q_ := \.$

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

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

* {{cite book , author=Dacorogna, B. , title=Introduction to the Calculus of Variations , publisher=Imperial College Press, London , year=2004 , isbn=1-86094-508-2

Geometry
Lipschitz maps
Sobolev spaces