Is the wikipedia definition of Lipschitz Euclidean domain correct?

See: http://en.wikipedia.org/wiki/Lipschitz_domain

i was wondering what stops me just showing the condition holds for one point and then just scale and translate that function $h_p$ for any point on the boundary... This doesn't seem right? What am I missing?

Thanks in advance.

  • 1
    What stops you is that the scalings and translations might not satisfy what they would need to. – user5810 Oct 29 '10 at 5:58
  • For each point p there is a corresponding radius r>0. Would I be allowed to scale and translate to x within the domain of h_p shrinking the radius about x appropriately? – alext87 Oct 29 '10 at 6:11
  • I can't come up with anything other than a circle that you could do that for. (try it with a corner and the midpoint of an edge from a square) – user5810 Oct 29 '10 at 6:24
  • I'm completely lost by the definition so I seem to get that it works. Why is it working for a circle? – alext87 Oct 29 '10 at 7:04
  • I am retagging this classical analysis. Euclidean domain does not mean what you think it means: en.wikipedia.org/wiki/Euclidean_domain – Willie Wong Oct 29 '10 at 9:06
up vote 7 down vote accepted

A domain of $\mathbb{R}^n$ with Lipschitz boundary is an open subset $\Omega\subset \mathbb{R}^n$, which is locally the sub-graph of a Lipschitz function w.r.to some choice of orthogonal coordinates. In other words, for any $p\in\partial \Omega$, up to an orthogonal change of coordinates, there is an open set $V\subset\mathbb{R}^{n-1}$, and a Lipschitz function $\phi:V\to(a,b)$ such that $U:=V\times (a,b)$ is a nbd of $p$ and $$\Omega\cap U=\{(x,t)\in U\ | \ t< \phi(x) \}.$$ This is equivalent to the definition given in the link, which is closer to the general definition of manifold with boundary (here the transition mappings are the bi-Lipschitz homeomorphisms).

In particular, for any point $p\in \partial\Omega$ there is a small cone $C$ with vertex in the origin, and a nbd $U$ of $p$ such that for any $q\in U\cap\partial\Omega$ the cone $q+C$ is disjoint from $\Omega$, and the cone $q-C$ is included in $\bar \Omega.$ This is a third equivalent definition.

The definition given in the link is sometimes called "weakly Lipschitz". The two definitions given by Pietro Majer are indeed equivalent, they are sometimes called "strongly Lipschitz" and "strong cone condition", respectively.

If the boundary of the domain is compact, then any cover of the boundary has a finite subcover. So in all the definitions, it doesn't matter whether we state the property for a neighborhood of every point, or for a finite cover.

One can easily see that strongly Lipschitz implies weakly Lipschitz. However, the converse is not true. This has to do with the fact that the implicit function theorem does not hold for Lipschitz continuous functions. E.g., there exist polyhedra that are are weakly Lipschitz but not strongly Lipschitz; they fail to be a graph of a Lipschitz continious function. For more see P. Grisvard, Elliptic Problems on Nonsmooth Domains, Pitman, Boston, London, Melbourne, 1985.

Unfortunately, in the literature "Lipschitz" is occasionally used meaning "strongly Lipschitz", but sometimes meaning "weakly Lipschitz".

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