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 Rⁿ. 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)) = Q₀;
• h_p(Ω ∩ B_r(p)) = Q₊;

where

$B_{r} (p) := \{ x \in \mathbb{R}^{n} | \| x - p \| < r \}$

denotes the n-dimensional open ball of radius r about p, Q denotes the unit ball B₁(0), and

$Q_{0} := \{ (x_{1}, \dots, x_{n}) \in Q | x_{n} = 0 \};$

$Q_{+} := \{ (x_{1}, \dots, x_{n}) \in Q | x_{n} > 0 \}.$

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.