Lipschitz domain

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 and bounded 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

: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 "B"1(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

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