On Laplacian eigenvalue equation with constant Neumann boundary data

TL;DR

This paper investigates boundary behaviors of solutions to the Laplacian eigenvalue problem with constant Neumann boundary data, deriving inequalities and symmetry results that characterize domain shapes like balls and rectangles.

Contribution

It introduces new inequalities involving eigenfunctions, characterizes domains where certain spectral constants are maximized, and extends symmetry breaking results to broader classes of domains.

Findings

Inequality for solutions on rectangular boxes, balls, and triangles with equality at symmetric domains.

Identification of conditions under which the Poincaré constant equals the second Neumann eigenvalue.

Extension of symmetry breaking results to wider classes of domains and quantitative estimates for thresholds.

Abstract

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.