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 Germany mathematician Rudolf Lipschitz.

Definition

Let [math]\displaystyle{ n \in \mathbb N }[/math]. Let [math]\displaystyle{ \Omega }[/math] be a domain of [math]\displaystyle{ \mathbb R^n }[/math] and let [math]\displaystyle{ \partial\Omega }[/math] denote the boundary of [math]\displaystyle{ \Omega }[/math]. Then [math]\displaystyle{ \Omega }[/math] is called a Lipschitz domain if for every point [math]\displaystyle{ p \in \partial\Omega }[/math] there exists a hyperplane [math]\displaystyle{ H }[/math] of dimension [math]\displaystyle{ n-1 }[/math] through [math]\displaystyle{ p }[/math], a Lipschitz-continuous function [math]\displaystyle{ g : H \rightarrow \mathbb R }[/math] over that hyperplane, and reals [math]\displaystyle{ r \gt 0 }[/math] and [math]\displaystyle{ h \gt 0 }[/math] such that

• [math]\displaystyle{ \Omega \cap C = \left\{x + y \vec{n} \mid x \in B_r(p) \cap H,\ -h \lt y \lt g(x) \right\} }[/math]
• [math]\displaystyle{ (\partial\Omega) \cap C = \left\{x + y \vec{n} \mid x \in B_r(p) \cap H,\ g(x) = y \right\} }[/math]

where

[math]\displaystyle{ \vec{n} }[/math] is a unit vector that is normal to [math]\displaystyle{ H, }[/math]

[math]\displaystyle{ B_{r} (p) := \{x \in \mathbb{R}^{n} \mid \| x - p \| \lt r \} }[/math] is the open ball of radius [math]\displaystyle{ r }[/math],

[math]\displaystyle{ C := \left\{x + y \vec{n} \mid x \in B_r(p) \cap H,\ -h \lt y \lt h \right\}. }[/math]

In other words, at each point of its boundary, [math]\displaystyle{ \Omega }[/math] is locally the set of points located above the graph of some Lipschitz function.

Generalization

A more general notion is that of weakly Lipschitz domains, which are domains whose boundary is locally flattable by a bilipschitz mapping. Lipschitz domains in the sense above are sometimes called strongly Lipschitz by contrast with weakly Lipschitz domains.

A domain [math]\displaystyle{ \Omega }[/math] is weakly Lipschitz if for every point [math]\displaystyle{ p \in \partial\Omega, }[/math] there exists a radius [math]\displaystyle{ r \gt 0 }[/math] and a map [math]\displaystyle{ l_p : B_r(p) \rightarrow Q }[/math] such that

• [math]\displaystyle{ l_p }[/math] is a bijection;
• [math]\displaystyle{ l_p }[/math] and [math]\displaystyle{ l_p^{-1} }[/math] are both Lipschitz continuous functions;
• [math]\displaystyle{ l_p\left( \partial\Omega \cap B_r(p) \right) = Q_0; }[/math]
• [math]\displaystyle{ l_p\left( \Omega \cap B_r(p) \right) = Q_+; }[/math]

where [math]\displaystyle{ Q }[/math] denotes the unit ball [math]\displaystyle{ B_1(0) }[/math] in [math]\displaystyle{ \mathbb{R}^{n} }[/math] and

[math]\displaystyle{ Q_{0} := \{(x_{1}, \ldots, x_{n}) \in Q \mid x_{n} = 0 \}; }[/math]

[math]\displaystyle{ Q_{+} := \{(x_{1}, \ldots, x_{n}) \in Q \mid x_{n} \gt 0 \}. }[/math]

A (strongly) Lipschitz domain is always a weakly Lipschitz domain but the converse is not true. An example of weakly Lipschitz domains that fails to be a strongly Lipschitz domain is given by the two-bricks domain

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.