Lipschitz domain

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.


Let n  N, and let Ω be an open subset of Rn. Let ∂Ω denote the 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 hp : Br(p)  Q such that


denotes the n-dimensional open ball of radius r about p, Q denotes the unit ball B1(0), and

Applications of Lipschitz domains

Many 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.


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