Description

• Title:
  Reverse Faber-Krahn inequalities for Zaremba problems
• Creator:
  Anoop, Thazhe Veetil;
  Ghosh, Mrityunjoy
• Description:
  Let $\Omega$ be a domain in $\mathbb{R}^n$ ($n\geq 2$) 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{alignat*}2 -\Delta_p u &=\tau \bigg(\int_{\Omega}|u|^q dx \bigg)^{({p-q})/{q}} |u|^{q-2}u &\quad&\text{in }\Omega,\\ u &=0&\quad&\text{on } \partial\Omega_D, \\ \frac{\partial u}{\partial \eta}&=0&\quad& \text{on } \partial \Omega\setminus \partial \Omega_D. \end{alignat*} Under the assumption that $\Omega_D$ is convex, we establish the following reverse Faber-Krahn inequality $$\tau_{1,q}(\Omega)\leq \tau_{1,q}({\Omega}^\star), $$% where ${\Omega}^\star=B_R\setminus \overline{B_r}$ is a concentric annular region in $\mathbb{R}^n$ having the same Lebesgue measure as $\Omega$ and such that \begin{enumerate}[(i)] \item (when $\Omega_D=\Omega_{\text{out}}$) $W_1(\Omega_D)= \omega_n R^{n-1}$, and $(\Omega^\star)_D=B_R$, \item (when $\Omega_D=\Omega_{\text{in}}$) $W_{n-1}(\Omega_D)=\omega_nr$, and $(\Omega^\star)_D=B_r$. \end{enumerate} Here $W_{i}(\Omega_D)$ is the $i^{\text{th}}$ {\it 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.
• Object availability:
  free access
• Rights:
  Prawa autorskie (c) 2024 Thazhe Veetil Anoop, Mrityunjoy Ghosh;
  https://creativecommons.org/licenses/by-nd/4.0
• Date:
  2024.09.15
• Type:
  info:eu-repo/semantics/article;
  info:eu-repo/semantics/publishedVersion;
  Recenzowany artykuł
• Language:
  English
• Source:
  Topological Methods in Nonlinear Analysis; Vol 64, No 1 (September 2024); 257 - 278;
  1230-3429
• Publisher:
  Nicolaus Copernicus University in Toruń
• Relation: https://apcz.umk.pl/TMNA/article/view/55210/39689 ;https://apcz.umk.pl/TMNA/article/view/55210/39690
• Subject:
  Zaremba problems;
  reverse Faber-Krahn inequality;
  Steiner formula;
  Nagy's inequality;
  method of interior parallel sets;
  35P15, 35P30, 49R05, 49Q10
• Identifier: https://apcz.umk.pl/TMNA/article/view/55210
• Data provider:
  Akademicka Platforma Czasopism
• Can I use it?:
  seek permission
• Type:
  other