Let $\Omega\subseteq\mathbb{R}^n$ be a domain and $\Sigma\subseteq \partial\Omega$ be a $C^1$, connected, open (in $\partial\Omega$) part of the boundary.
Does there always exist a subdomain $\Omega'\subseteq \Omega$ with $C^1$ boundary such that $\Sigma\subseteq \partial\Omega'$ ?
I'm trying to use Green representation formula with such subdomain $\Omega'$.