Reverse Faber-Krahn inequalities for Zaremba problems
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| Format: | Preprint |
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2022
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| author | Anoop, T. V. Ghosh, Mrityunjoy |
| author_facet | Anoop, T. V. Ghosh, Mrityunjoy |
| contents | Let $Ω$ be a multiply-connected domain in $\mathbb{R}^n$ ($n\geq 2$) of the form $Ω=Ω_{\text{out}}\setminus \bar{Ω_{\text{in}}}.$ Set $Ω_D$ to be either $Ω_{\text{out}}$ or $Ω_{\text{in}}$. For $p\in (1,\infty),$ and $q\in [1,p],$ let $τ_{1,q}(Ω)$ 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 $Ω_D$ is convex, we establish the following reverse Faber-Krahn inequality $$τ_{1,q}(Ω)\leq τ_{1,q}(Ω^\bigstar),$$ where $Ω^\bigstar=B_R\setminus \bar{B_r}$ is a concentric annular region in $\mathbb{R}^n$ having the same Lebesgue measure as $Ω$ and such that (i) (when $Ω_D=Ω_{\text{out}}$) $W_1(Ω_D)= ω_n R^{n-1}$, and $(Ω^\bigstar)_D=B_R$, (ii) (when $Ω_D=Ω_{\text{in}}$) $W_{n-1}(Ω_D)=ω_nr$, and $(Ω^\bigstar)_D=B_r$.
Here $W_{i}(Ω_D)$ is the $i^{\text{th}}$ $quermassintegral$ of $Ω_D.$ 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2205_12717 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Reverse Faber-Krahn inequalities for Zaremba problems Anoop, T. V. Ghosh, Mrityunjoy Analysis of PDEs Optimization and Control 35P15, 35P30, 49R05, 49Q10 Let $Ω$ be a multiply-connected domain in $\mathbb{R}^n$ ($n\geq 2$) of the form $Ω=Ω_{\text{out}}\setminus \bar{Ω_{\text{in}}}.$ Set $Ω_D$ to be either $Ω_{\text{out}}$ or $Ω_{\text{in}}$. For $p\in (1,\infty),$ and $q\in [1,p],$ let $τ_{1,q}(Ω)$ 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 $Ω_D$ is convex, we establish the following reverse Faber-Krahn inequality $$τ_{1,q}(Ω)\leq τ_{1,q}(Ω^\bigstar),$$ where $Ω^\bigstar=B_R\setminus \bar{B_r}$ is a concentric annular region in $\mathbb{R}^n$ having the same Lebesgue measure as $Ω$ and such that (i) (when $Ω_D=Ω_{\text{out}}$) $W_1(Ω_D)= ω_n R^{n-1}$, and $(Ω^\bigstar)_D=B_R$, (ii) (when $Ω_D=Ω_{\text{in}}$) $W_{n-1}(Ω_D)=ω_nr$, and $(Ω^\bigstar)_D=B_r$. Here $W_{i}(Ω_D)$ is the $i^{\text{th}}$ $quermassintegral$ of $Ω_D.$ 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. |
| title | Reverse Faber-Krahn inequalities for Zaremba problems |
| topic | Analysis of PDEs Optimization and Control 35P15, 35P30, 49R05, 49Q10 |
| url | https://arxiv.org/abs/2205.12717 |