Lipschitz boundary

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

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