Lipschitz domain explained

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 \in N . Let \Omega be a domain of R ^{n} and let \partial\Omega denote the boundary of \Omega . Then \Omega is called a Lipschitz domain if for every point p \in \partial\Omega there exists a hyperplane H of dimension n -1 through p , a Lipschitz-continuous function g : H → R over that hyperplane, and reals r > 0 and h > 0 such that\Omega \cap C = \left\{x + y \vec{n} \mid x \in B _{r(p)} \cap H, - h < y < g(x) \right\} (\partial\Omega) \cap C = \left\{x + y \vec{n} \mid x \in B _{r(p)} \cap H, g(x) = y \right\} where \vec{n} is a unit vector that is normal to H, B _{r} (p) : = \{x \in R ^{n} \mid \| x - p \| < r \} is the open ball of radius r ,C : = \left\{x + y \vec{n} \mid x \in B _{r(p)} \cap H, { - h} < y < h \right\}.

In other words, at each point of its boundary,

\Omega 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

\Omega is weakly Lipschitz if for every point p \in \partial\Omega, there exists a radius r > 0 and a map \ell _{p} : B _{r(p)} → Q such that\ell _{p} is a bijection;\ell _{p} and l ^{-1} _{p} are both Lipschitz continuous functions;\ell _{p\left(} \partial\Omega \cap B _{r(p)} \right) = Q _{0;} \ell _{p\left(} \Omega \cap B _{r(p)} \right) = Q _{+;} where Q denotes the unit ball B _{1(0)} in R ^{n} andQ _{0} : = \{(x _{1,} \ldots, x _{n)} \in Q \mid x _{n} = 0 \}; Q _{+} : = \{(x _{1,} \ldots, x _{n)} \in Q \mid x _{n} > 0 \}.

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 ^[1]

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

• Book: Dacorogna, B. . Introduction to the Calculus of Variations . Imperial College Press, London . 2004 . 1-86094-508-2 .

Notes and References

1. Werner Licht, M. "Smoothed Projections over Weakly Lipschitz Domains", arXiv, 2016.