I don't quite understand how to apply the definition (Understanding Lipschitz domain) of Lipschitz domain. My question is about annulus (of which punctured disc is a special case). Is annulus a Lipschitz domain. How can we write its boundary(two disjoint circles) as the graph of a Lipschitz map locally? I know we can cover these circles by charts locally. 
2026-04-01 12:08:23.1775045303
Is punctured disc a Lipschitz domain?
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The annulus, i.e., $$ A_{r,R}=\{(x,y):r^2<x^2+y^2<R^2\} $$ where $0<r<R$ is a Lipschitz domain, since its boundary is the level set of a Lipschitz continuous function.
However, for $R>r=0$, the punctured disc $$ A=\{(x,y):0<x^2+y^2<R^2\} $$ is not a Lipschitz domain.