Lipschitz domain

Let n \in \mathbb N. Let \Omega be a domain of \mathbb 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 \rightarrow \mathbb 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 • (\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 one of the two unit vectors that are normal to H, :B_{r} (p) := \{x \in \mathbb{R}^{n} \mid \| x - p \| is the open ball of radius r, :C := \left\{x + y \vec{n} \mid x \in B_r(p) \cap H,\ {-h} 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) \rightarrow Q such that • \ell_p is a bijection; • \ell_p and \ell_p^{-1} 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 \mathbb{R}^n and :Q_{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 ==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==