A Lipschitz domain $\Omega$ is a domain in $\mathbb R^n$ whose boundary $\partial\Omega$ is locally the graph of a Lipschitz continuous function. For example, any $C^1$ domain is Lipschitz and a cube is Lipschitz but not $C^1$.
An important parameter of a Lipschitz domain is its Lipschitz constant $\tau$. Its arctangent ($\arctan \tau$) corresponds to the maximum angle that the boundary of the domain can make. Thus the Lipschitz constant of the cube is $1$ but the Lipschitz constant of a $C^1$ domain can be chosen arbitrarily small. One of the interesting things of Lipschitz domains is that their regularity is "scale-invariant". A cube does not get "softer" after zooming in (at least when you zoom in at a corner).
I'd like to know:
- pathologic examples of Lipschitz domains, specially related with harmonic functions defined inside (or other topics like harmonic analysis).
- things that stop working when you go from $C^1$ domains to Lipschitz domains or that get much harder (for example, I know that the theory of layer potentials gets harder).
- also I'm curious whether given a point $x\in\partial\Omega$ we can find an open set $U \subset \partial\Omega$ such that $U$ is star-shaped with respect to $x$. Or if that's not possible, whether it's possible a.e. w.r.t. surface measure.