I encountered a problem which leads to prove $\frac{1}{\left | \varphi _{1}(x) \right |}$ is a bounded function, $ \varphi _{1}$ is the first eigenfunction of laplacian in an open, bounded, smooth domain in $\mathbb{R}^n$, namely $\Omega$. Since the closure of $\Omega$ is compact, I'm hoping on the continuity of $ \varphi _{1}$. But I'm not sure if $ \varphi _{1}$ is in the class $C^1(\overline{\Omega} )$?
If possible please give me some book titles or lecture notes about this aspect (I'm a beginner).
Thank you.