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When do we call the boundary $\partial\Omega$ of a bounded domain $\Omega\subseteq\mathbb{R}^n$ smooth? I can't find a formal definition.

I know, that we say, that $\partial\Omega$ has a $C^k$-boundary ($\partial\Omega\in C^k$), if it can be locally represented by the graph of a $C^k$-function. So, does "$\partial\Omega$ is smooth" mean, that $\partial\Omega\in C^\infty$?

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  • $\begingroup$ I think it means that it can be represented locally as graph of a $C^\infty$-function. $\endgroup$ – user99914 Jun 27 '15 at 11:32
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Go to check Leoni's book, section 12.1 about extension domains, in which he carefully defined the Lipschitz boundary. For example Definition 12.9 and 12.10. Once you understand the Lipschitz boundary, the other boundary, like smooth boundary you mentioned, can be defined similarly.

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