Let . Let be a domain of and let denote the boundary of . Then is called a Lipschitz domain if for every point there exists a hyperplane of dimension through , a Lipschitz-continuous function over that hyperplane, and reals and such that
In other words, at each point of its boundary, is locally the set of points located above the graph of some Lipschitz function.
Generalizationedit
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 is weakly Lipschitz if for every point there exists a radius and a map such that
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 [1]
^Werner Licht, M. "Smoothed Projections over Weakly Lipschitz Domains", arXiv, 2016.
Dacorogna, B. (2004). Introduction to the Calculus of Variations. Imperial College Press, London. ISBN1-86094-508-2.
January 01, 1970
lipschitz, domain, mathematics, domain, with, lipschitz, boundary, domain, euclidean, space, whose, boundary, sufficiently, regular, sense, that, thought, locally, being, graph, lipschitz, continuous, function, term, named, after, german, mathematician, rudolf. 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 Contents 1 Definition 2 Generalization 3 Applications of Lipschitz domains 4 ReferencesDefinition editLet n N displaystyle n in mathbb N nbsp Let W displaystyle Omega nbsp be a domain of R n displaystyle mathbb R n nbsp and let W displaystyle partial Omega nbsp denote the boundary of W displaystyle Omega nbsp Then W displaystyle Omega nbsp is called a Lipschitz domain if for every point p W displaystyle p in partial Omega nbsp there exists a hyperplane H displaystyle H nbsp of dimension n 1 displaystyle n 1 nbsp through p displaystyle p nbsp a Lipschitz continuous function g H R displaystyle g H rightarrow mathbb R nbsp over that hyperplane and reals r gt 0 displaystyle r gt 0 nbsp and h gt 0 displaystyle h gt 0 nbsp such that W C x y n x B r p H h lt y lt g x displaystyle Omega cap C left x y vec n mid x in B r p cap H h lt y lt g x right nbsp W C x y n x B r p H g x y displaystyle partial Omega cap C left x y vec n mid x in B r p cap H g x y right nbsp where n displaystyle vec n nbsp is a unit vector that is normal to H displaystyle H nbsp B r p x R n x p lt r displaystyle B r p x in mathbb R n mid x p lt r nbsp is the open ball of radius r displaystyle r nbsp C x y n x B r p H h lt y lt h displaystyle C left x y vec n mid x in B r p cap H h lt y lt h right nbsp In other words at each point of its boundary W displaystyle Omega nbsp is locally the set of points located above the graph of some Lipschitz function Generalization editA 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 W displaystyle Omega nbsp is weakly Lipschitz if for every point p W displaystyle p in partial Omega nbsp there exists a radius r gt 0 displaystyle r gt 0 nbsp and a map l p B r p Q displaystyle l p B r p rightarrow Q nbsp such that l p displaystyle l p nbsp is a bijection l p displaystyle l p nbsp and l p 1 displaystyle l p 1 nbsp are both Lipschitz continuous functions l p W B r p Q 0 displaystyle l p left partial Omega cap B r p right Q 0 nbsp l p W B r p Q displaystyle l p left Omega cap B r p right Q nbsp where Q displaystyle Q nbsp denotes the unit ball B 1 0 displaystyle B 1 0 nbsp in R n displaystyle mathbb R n nbsp and Q 0 x 1 x n Q x n 0 displaystyle Q 0 x 1 ldots x n in Q mid x n 0 nbsp Q x 1 x n Q x n gt 0 displaystyle Q x 1 ldots x n in Q mid x n gt 0 nbsp 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 1 Applications of Lipschitz domains editMany 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 edit Werner Licht M Smoothed Projections over Weakly Lipschitz Domains arXiv 2016 Dacorogna B 2004 Introduction to the Calculus of Variations Imperial College Press London ISBN 1 86094 508 2 Retrieved from https en wikipedia org w index php title Lipschitz domain amp oldid 1179449726, wikipedia, wiki, book, books, library,