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In
mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ...
, a Lipschitz domain (or domain with Lipschitz boundary) is a
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
in
Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ...
whose boundary is "sufficiently regular" in the sense that it can be thought of as locally being the graph of a . The term is named after the
German German(s) may refer to: Common uses * of or related to Germany * Germans, Germanic ethnic group, citizens of Germany or people of German ancestry * For citizens of Germany, see also German nationality law * German language The German la ...

mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces ...

Rudolf Lipschitz Rudolf Otto Sigismund Lipschitz (14 May 1832 β 7 October 1903) was a German mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such top ...
.

Definition

Let $n \in \mathbb N$. Let $\Omega$ be a
domain Domain may refer to: Mathematics *Domain of a function In mathematics, the domain of a Function (mathematics), function is the Set (mathematics), set of inputs accepted by the function. It is sometimes denoted by \operatorname(f), where is th ...
of $\mathbb R^n$ and let $\partial\Omega$ denote the
boundary Boundary or Boundaries may refer to: * Border, in political geography Entertainment *Boundaries (2016 film), ''Boundaries'' (2016 film), a 2016 Canadian film *Boundaries (2018 film), ''Boundaries'' (2018 film), a 2018 American-Canadian road trip ...
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\$ * $\left(\partial\Omega\right) \cap C = \left\$ where :$\vec$ is a unit vector that is normal to $H,$ :$B_ \left(p\right) := \$ 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 $l_p : B_r\left(p\right) \rightarrow Q$ such that * $l_p$ is a
bijection In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

; * $l_p$ and $l_p^$ are both Lipschitz continuous functions; * $l_p\left\left( \partial\Omega \cap B_r\left(p\right) \right\right) = Q_0;$ * $l_p\left\left( \Omega \cap B_r\left(p\right) \right\right) = Q_+;$ where $Q$ denotes the unit ball $B_1\left(0\right)$ 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 arXiv (pronounced "archive"βthe X represents the Chi (letter), Greek letter chi  is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not Scholarly peer review, ...
'', 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 equation In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...
s 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