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.

Such domains are also called strongly Lipschitz domains to contrast them with weakly Lipschitz domains, which are a more general class of domains. A weakly Lipschitz domain is a domain whose boundary is locally flattable by a Lipschitzeomorphism.

Definition

Let . Let be an open subset 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 the values and such that

where

is a unit vector that is normal to

More generally, is said to be 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

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.

References

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