Reverse Faber-Krahn inequalities for Zaremba problems

TL;DR

This paper proves reverse Faber-Krahn inequalities for Zaremba eigenvalue problems on multiply-connected convex domains, showing that certain annular regions maximize the first eigenvalue under measure constraints.

Contribution

It introduces reverse Faber-Krahn inequalities for Zaremba eigenvalues on convex multiply-connected domains, extending classical spectral inequalities to these settings.

Findings

Eigenvalue bounds for annular regions match measure constraints.

Established inequalities for convex domains with specific boundary conditions.

Extended results to parallel sets of convex domains in higher dimensions.

Abstract

Let $\Omega$ be a multiply-connected domain in $\mathbb{R}^n$ ($n\geq 2$) of the form $\Omega=\Omega_{\text{out}}\setminus \bar{\Omega_{\text{in}}}.$ Set $\Omega_D$ to be either $\Omega_{\text{out}}$ or $\Omega_{\text{in}}$. For $p\in (1,\infty),$ and $q\in [1,p],$ let $\tau_{1,q}(\Omega)$ be the first eigenvalue of \begin{equation*} -\Delta_p u =\tau \left(\int_{\Omega}|u|^q \text{d}x \right)^{\frac{p-q}{q}} |u|^{q-2}u\;\text{in} \;\Omega,\; u =0\;\text{on}\;\partial\Omega_D, \frac{\partial u}{\partial \eta}=0\;\text{on}\; \partial \Omega\setminus \partial \Omega_D. \end{equation*} Under the assumption that $\Omega_D$ is convex, we establish the following reverse Faber-Krahn inequality $$\tau_{1,q}(\Omega)\leq \tau_{1,q}({\Omega}^\bigstar),$$ where ${\Omega}^\bigstar=B_R\setminus \bar{B_r}$ is a concentric annular region in $\mathbb{R}^n$ having the same Lebesgue measure as $\Omega$ and such that (i) (when $\Omega_D=\Omega_{\text{out}}$) $W_1(\Omega_D)= \omega_n R^{n-1}$, and $(\Omega^\bigstar)_D=B_R$, (ii) (when $\Omega_D=\Omega_{\text{in}}$) $W_{n-1}(\Omega_D)=\omega_nr$, and $(\Omega^\bigstar)_D=B_r$. Here $W_{i}(\Omega_D)$ is the $i^{\text{th}}$ $quermassintegral$ of $\Omega_D.$ We also establish Sz. Nagy's type inequalities for parallel sets of a convex domain in $\mathbb{R}^n$ ($n\geq 3$) for our proof.