Let $\varphi_1$ be the first eigenfunction of the zero Dirichlet Laplacian in a planar bounded domain $\Omega$. That is, $$ \left\{ \begin{aligned} -\Delta \varphi_1 &= \lambda_1 \varphi_1 &&\text{in } \Omega \subset \mathbb{R}^2,\\ \varphi_1 &= 0 &&\text{on } \partial \Omega. \end{aligned} \right. $$
If $\Omega$ has corners, and we look at the plot of $\varphi_1$, then we see that its normal derivative tends to zero near exterior (outward) corners, and tends to infinity near interior (inward) corners.
This fact suggests that, in a smooth domain, there should be some connection between curvature of the boundary at a point and the normal derivative of $\varphi_1$ at this point. That is, if the curvature is big positive, then the normal derivative is small. And if the curvature is big negative, then the normal derivative is large.
However, I was not able to find corresponding inequalities in the literature. I would appreciate some references to such facts and related results in this direction.
Thanks!