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.
denote the boundary of
is called a Lipschitz domain if for every point
there exists a hyperplane
such that where In other words, at each point of its boundary,
is locally the set of points located above the graph of some Lipschitz function.
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.
is weakly Lipschitz if for every point
denotes the unit ball
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 [1] 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.