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mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, a Lipschitz domain (or domain with Lipschitz boundary) is a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
in
Euclidean space Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, that is, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean ...
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 A mathematician is someone who uses an extensive knowledge of mathematics in their work, typically to solve mathematical problems. Mathematicians are concerned with numbers, data, quantity, structure, space, models, and change. History ...
Rudolf Lipschitz Rudolf Otto Sigismund Lipschitz (14 May 1832 – 7 October 1903) was a German mathematician who made contributions to mathematical analysis (where he gave his name to the Lipschitz continuity condition) and differential geometry, as well as numbe ...
.


Definition

Let n \in \mathbb N. Let \Omega be a
domain Domain may refer to: Mathematics *Domain of a function, the set of input values for which the (total) function is defined ** Domain of definition of a partial function ** Natural domain of a partial function ** Domain of holomorphy of a function * ...
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), a 2016 Canadian film * ''Boundaries'' (2018 film), a 2018 American-Canadian road trip film *Boundary (cricket), the edge of the pla ...
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 In mathematics, a bijection, also known as a bijective function, one-to-one correspondence, or invertible function, is a function between the elements of two sets, where each element of one set is paired with exactly one element of the other s ...
; * 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 arXiv (pronounced "archive"—the X represents the Greek letter chi ⟨χ⟩) is an open-access repository of electronic preprints and postprints (known as e-prints) approved for posting after moderation, but not peer review. It consists of ...
'', 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, a partial differential equation (PDE) is an equation which imposes relations between the various partial derivatives of a multivariable function. The function is often thought of as an "unknown" to be solved for, similarly to h ...
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