In

"Smoothed Projections over Weakly Lipschitz Domains"

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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
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in Euclidean space
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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
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Rudolf Lipschitz
Rudolf Otto Sigismund Lipschitz (14 May 1832 β 7 October 1903) was a German mathematician
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.
Definition

Let $n\; \backslash in\; \backslash mathbb\; N$. Let $\backslash Omega$ be adomain
Domain may refer to:
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*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 $\backslash mathbb\; R^n$ and let $\backslash partial\backslash Omega$ denote the boundary
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of $\backslash Omega$. Then $\backslash Omega$ is called a Lipschitz domain if for every point $p\; \backslash in\; \backslash partial\backslash Omega$ there exists a hyperplane $H$ of dimension $n-1$ through $p$, a Lipschitz-continuous function $g\; :\; H\; \backslash rightarrow\; \backslash mathbb\; R$ over that hyperplane, and reals $r\; >\; 0$ and $h\; >\; 0$ such that
* $\backslash Omega\; \backslash cap\; C\; =\; \backslash left\backslash $
* $(\backslash partial\backslash Omega)\; \backslash cap\; C\; =\; \backslash left\backslash $
where
:$\backslash vec$ is a unit vector that is normal to $H,$
:$B\_\; (p)\; :=\; \backslash $ is the open ball of radius $r$,
:$C\; :=\; \backslash left\backslash .$
In other words, at each point of its boundary, $\backslash Omega$ 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 $\backslash Omega$ is weakly Lipschitz if for every point $p\; \backslash in\; \backslash partial\backslash Omega,$ there exists a radius $r\; >\; 0$ and a map $l\_p\; :\; B\_r(p)\; \backslash rightarrow\; Q$ such that * $l\_p$ is abijection
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;
* $l\_p\backslash left(\; \backslash partial\backslash Omega\; \backslash cap\; B\_r(p)\; \backslash right)\; =\; Q\_0;$
* $l\_p\backslash left(\; \backslash Omega\; \backslash cap\; B\_r(p)\; \backslash right)\; =\; Q\_+;$
where $Q$ denotes the unit ball $B\_1(0)$ in $\backslash mathbb^$ and
:$Q\_\; :=\; \backslash ;$
:$Q\_\; :=\; \backslash .$
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"

'''', 2016.

Applications of Lipschitz domains

Many of the Sobolev embedding theorems require that the domain of study be a Lipschitz domain. Consequently, manypartial 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