Disclaimer. I posted this question in Math.SE, but it haven't received enough attention.

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 $\Omega$ is piece-wise smooth and 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.

(See the plot for the standard L-shape here:  https://i.sstatic.net/j28X4.png I don't have enough reputation to post images.)

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 close to zero. 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. (Although I believe that such results should be well-known.) I would appreciate some references to such facts and related results in this direction.

Thanks!

Disclaimer. I posted this question in Math.SE, but it haven't received enough attention.

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 $\Omega$ is piece-wise smooth and 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.

See the plot for the standard L-shape: 

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 close to zero. 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. (Although I believe that such results should be well-known.) I would appreciate some references to such facts and related results in this direction.

Thanks!