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. 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. Such domains are also called strongly Lipschitz domains to contrast them with weakly Lipschitz domains, which are a more general class of domains. A weakly Lipschitz domain is a domain whose boundary is locally flattable by a Lipschitzeomorphism. Let n ∈ N {displaystyle nin mathbb {N} } . Let Ω {displaystyle Omega } be an open subset of R n {displaystyle mathbb {R} ^{n}} and let ∂ Ω {displaystyle partial Omega } denote the boundary of Ω {displaystyle Omega } . Then Ω {displaystyle Omega } is called a Lipschitz domain if for every point p ∈ ∂ Ω {displaystyle pin partial Omega } there exists a hyperplane H {displaystyle H} of dimension n − 1 {displaystyle n-1} through p {displaystyle p} , a Lipschitz-continuous function g : H → R {displaystyle g:H ightarrow mathbb {R} } over that hyperplane, and the values r > 0 {displaystyle r>0} and h > 0 {displaystyle h>0} such that