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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.

Contents

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 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 ∈ 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 , … , x n ) ∈ Q | x n = 0 } ; Q + := { ( x 1 , … , x n ) ∈ 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

    Lipschitz domain Wikipedia
    (Text) CC BY-SA