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.

Definition

Let. Let \Omega be a domain of \mathbb R^n and let denote the boundary of \Omega. Then \Omega is called a Lipschitz domain if for every point there exists a hyperplane H of dimension n-1 through p, a Lipschitz-continuous function over that hyperplane, and reals r > 0 and h > 0 such that where 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 there exists a radius r > 0 and a map such that where Q denotes the unit ball B_1(0) in and 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

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.