On Laplacian eigenvalue equation with constant Neumann boundary data

Let \(\Omega \) $\mathrm{\Omega }$ be a bounded Lipschitz domain in \(\mathbb {R}^n\) ${\mathbb{R}}^n$ and we study boundary behaviors of solutions to the following Laplacian eigenvalue equation with constant Neumann data: 0.1 \(\begin{aligned} {\left\{ \begin{array}{ll} -\Delta u=cu\quad &{}\text{ in }\, \Omega \\ \frac{\partial u}{\partial \nu }=-1\quad &{}\text{ on }\, \partial \Omega . \end{array}\right. } \end{aligned}\) $$\begin{array}{c}\hfill \left\{\begin{array}{cc}-\mathrm{\Delta }u=cu\hfill & \text{in}\mathrm{\Omega }\hfill \\ \frac{\partial u}{\partial \nu }=-1\hfill & \text{on}\partial \mathrm{\Omega }.\hfill \end{array}\right)\end{array}$$ 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: 0.2 \(\begin{aligned} \lim _{c\rightarrow \mu _2^-}c\int _{\partial \Omega }u_c\, \textrm{d} \sigma \ge \frac{\hbox {n}-1}{\hbox {n}}\frac{\hbox {P}^2(\Omega )}{|\Omega |}, \end{aligned}\) $$\begin{array}{c}\hfill \underset{c\to {\mu }_2^{-}}{lim}c{\int }_{\partial \mathrm{\Omega }}{u}_c\text{d}\sigma \ge \frac{\text{n}-1}{\text{n}}\frac{{\text{P}}^2(\mathrm{\Omega })}{|\mathrm{\Omega }|},\end{array}$$ with equality achieved only at cubes and balls. In the above, \(u_c\) $u_c$ is the solution to (0.1) and \(\mu _2\) ${\mu }_2$ is the second Neumann Laplacian eigenvalue. Second, let \(\kappa _1\) ${\kappa }_1$ be the best constant for the Poincaré inequality with the vanishing mean condition over \(\partial \Omega \) $\partial \mathrm{\Omega }$ , and we prove that \(\kappa _1\le \mu _2\) ${\kappa }_1\le {\mu }_2$ and that the equality holds if and only if \(\int _{\partial \Omega }u_c\, \textrm{d} \sigma >0\) ${\int }_{\partial \mathrm{\Omega }}{u}_c\text{d}\sigma >0$ for any \(c\in (0,\mu _2)\) $c\in (0,{\mu }_2)$ . As a consequence, \(\kappa _1=\mu _2\) ${\kappa }_1={\mu }_2$ on balls, rectangular boxes and equilateral triangles, and balls maximize \(\kappa _1\) ${\kappa }_1$ 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 (9) 107(4): 451–463, 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 \(\kappa _1<\mu _2\) ${\kappa }_1<{\mu }_2$ , (0.2) is never true, while whether it is valid for domains on which \(\kappa _1=\mu _2\) ${\kappa }_1={\mu }_2$ remains open.