Reverse Faber-Krahn inequalities for Zaremba problems

Abstract

Let $\Omega$ be a multiply-connected domain in ${\mathbb{R}}^n$ ($n\ge 2$) of the form $\Omega ={\Omega }_{\text{out}}\setminus \overline{{\Omega }_{\text{in}}}.$ 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*} -Δ_p u =τ\left(\int_Ω|u|^q \text{d}x \right)^{\frac{p-q}{q}} |u|^{q-2}u\;\text{in} \;Ω,\; u =0\;\text{on}\;\partialΩ_D, \frac{\partial u}{\partial η}=0\;\text{on}\; \partial Ω\setminus \partial Ω_D. \end{equation*} Under the assumption that ${\Omega }_D$ is convex, we establish the following reverse Faber-Krahn inequality $${\tau }_{1,q}(\Omega )\le {\tau }_{1,q}({\Omega }^{★}),$$ where ${\Omega }^{★}=B_R\setminus \overline{B_r}$ 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 $({\Omega }^{★}{)}_D=B_R$, (ii) (when ${\Omega }_D={\Omega }_{\text{in}}$) $W_{n-1}({\Omega }_D)={\omega }_{n}r$, and $({\Omega }^{★}{)}_D=B_r$. 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 ${\mathbb{R}}^n$ ($n\ge 3$) for our proof.

17 pages; V2; Minor changes are made in the statements and proofs of equality case