Consider the following problem, where $\Omega$ is a domain having a $C^1$ boundary in the neighborhood of $x_2$. Let $B$ be the green ball with radius $\varepsilon_2$. Now consider the point $x_2 \in \partial B \cap \partial \Omega$.
In a neighborhood of $x_2$ both boundaries ($\partial\Omega$ and $\partial B$) have an outer normal field $\nu_\Omega$ resp. $\nu_B$. Does $$ \nu_B(x_2) = \nu_\Omega(x_2) $$ hold?
Is it possible to further generalize this fact to the case that $\Omega_1$ and $\Omega_2$ are domains with $C^1$ boundary under the assumption that $\Omega_1 \subseteq \Omega_2$ and $\partial\Omega_1 \cap \partial\Omega_2 \neq \emptyset$?