Lipschitz domain

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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 n ∈ N, and let Ω be an open and bounded 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 hp : Br(p) → Q such that

  • hp is a bijection;
  • hp and hp−1 are both Lipschitz continuous functions;
  • hp(∂Ω ∩ Br(p)) = Q0;
  • hp(Ω ∩ Br(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[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.