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
is a unit vector that is normal to
is the open ball of radius ,
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.
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