  Let $\Omega$ be a bounded Lipshcitz domain in $\mathbb{R}^n$ and we study
boundary behaviors of solutions to the Laplacian eigenvalue equation with
constant Neumann data. \begin{align}
  \label{cequation0} \begin{cases} -\Delta u=cu\quad &\mbox{in $\Omega$}\\
\frac{\partial u}{\partial \nu}=-1\quad &\mbox{on $\partial \Omega$}.
\end{cases} \end{align}First, by using properties of Bessel functions and
proving new inequalities on elementary symmetric polynomials, we obtain the
following inequality for rectangular boxes, balls and equilateral triangles:
\begin{align} \label{bbb}
  \lim_{c\rightarrow \mu_2^-}c\int_{\partial \Omega}u_c\, d\sigma\ge
\frac{n-1}{n}\frac{P^2(\Omega)}{|\Omega|}, \end{align}with equality achieved
only at cubes and balls. In the above, $u_c$ is the solution to the eigenvalue
equation and $\mu_2$ is the second Neumann Laplacian eigenvalue. Second, let
$\kappa_1$ be the best constant for the Poincar\'e inequality with mean zero on
$\partial \Omega$, and we prove that $\kappa_1\le \mu_2$, with equality holds
if and only if $\int_{\partial \Omega}u_c\, d\sigma>0$ for any $c\in
(0,\mu_2)$. As a consequence, $\kappa_1=\mu_2$ on balls, rectangular boxes and
equilateral triangles, and balls maximize $\kappa_1$ over all Lipschitz domains
with fixed volume. As an application, we extend the symmetry breaking results
from ball domains obtained in Bucur-Buttazzo-Nitsch[J. Math. Pures Appl.,
2017], to wider class of domains, and give quantitative estimates for the
precise breaking threshold at balls and rectangular boxes. It is a direct
consequence that for domains with $\kappa_1<\mu_2$, the above boundary limit
inequality is never true, while whether it is valid for domains on which
$\kappa_1=\mu_2$ remains open.
