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2012-05-08T11:46:39Z

ISBNs (Build KF)

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In [[mathematics]], a '''Lipschitz domain''' (or '''domain with Lipschitz boundary''') is a [[Domain (mathematical analysis)|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 continuity|Lipschitz continuous function]]. The term is named after the [[Germany|German]] [[mathematician]] [[Rudolf Lipschitz]].

==Definition==

Let ''n'' ∈ '''N''', and let Ω be an [[open set|open]] and [[bounded set|bounded]] [[subset]] of '''R'''''n''. Let ∂Ω denote the [[boundary (topology)|boundary]] of Ω. Then Ω is said to have '''Lipschitz boundary''', and is called a '''Lipschitz domain''', if, for every point ''p'' ∈ ∂Ω, there exists a radius ''r'' > 0 and a map ''h''''p'' : ''B''''r''(''p'') → ''Q'' such that
* ''h''''p'' is a [[bijection]];
* ''h''''p'' and ''h''''p''−1 are both Lipschitz continuous functions;
* ''h''''p''(∂Ω ∩ ''B''''r''(''p'')) = ''Q''0;
* ''h''''p''(Ω ∩ ''B''''r''(''p'')) = ''Q''+;
where

:<math>B_{r} (p) := \{ x \in \mathbb{R}^{n} | \| x - p \| < r \}</math>

denotes the ''n''-[[dimension]]al [[open ball]] of radius ''r'' about ''p'', ''Q'' denotes the unit ball ''B''1(0), and

:<math>Q_{0} := \{ (x_{1}, \dots, x_{n}) \in Q | x_{n} = 0 \};</math>
:<math>Q_{+} := \{ (x_{1}, \dots, x_{n}) \in Q | x_{n} > 0 \}.</math>

==Applications of Lipschitz domains==

Many of the [[Sobolev inequality|Sobolev embedding theorems]] require that the domain of study be a Lipschitz domain. Consequently, many [[partial differential equation]]s and [[calculus of variations|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 }}

[[Category:Geometry]]
[[Category:Lipschitz maps]]
[[Category:Sobolev spaces]]

Nice name - Revision history

en>Helpful Pixie Bot: ISBNs (Build KF)