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.
Definition
Let n ∈ N, and let Ω be an open subset of R^{n}. Let ∂Ω denote the boundary of Ω. Then Ω is said to have Lipschitz boundary, and is called a Lipschitz domain, if, for every point p ∈ ∂Ω, there exists a radius r > 0 and a map h_{p} : B_{r}(p) → Q such that
- h_{p} is a bijection;
- h_{p} and h_{p}^{−1} are both Lipschitz continuous functions;
- h_{p}(∂Ω ∩ B_{r}(p)) = Q_{0};
- h_{p}(Ω ∩ B_{r}(p)) = Q_{+};
where
denotes the n-dimensional open ball of radius r about p, Q denotes the unit ball B_{1}(0), 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.