# 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.