Lipschitz domain

(Redirected from Lipschitz boundary)

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.