Lipschitz domain

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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[edit]

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

where

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.

Generalization[edit]

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

  • is a bijection;
  • and are both Lipschitz continuous functions;

where denotes the unit ball 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 [1]


Applications of Lipschitz domains[edit]

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.

References[edit]

  • Dacorogna, B. (2004). Introduction to the Calculus of Variations. Imperial College Press, London. ISBN 1-86094-508-2.