Anoop, Thazhe Veetil; Ghosh, Mrityunjoy Reverse Faber-Krahn inequalities for Zaremba problems. (English) Zbl 07959970 Topol. Methods Nonlinear Anal. 64, No. 1, 257-278 (2024). Summary: 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 \[ \begin{aligned} -\Delta_p u &=\tau \bigg(\int_{\Omega}|u|^q dx \bigg)^{(p-q)/q} |u|^{q-2}u & &\text{in }\Omega, \\ u &=0 \quad & & \text{on } \partial\Omega_D,\\ \dfrac{\partial u}{\partial \eta}&=0 \quad& &\text{on } \partial \Omega\setminus \partial \Omega_D. \end{aligned} \] 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 (i) (when \(\Omega_D =\Omega_{\text{out}}\)) \(W_1 (\Omega_D)= \omega_n R^{n-1}\), and \((\Omega^{\star})_D =B_R\),(ii) (when \(\Omega_D =\Omega_{\text{in}}\)) \(W_{n-1}(\Omega_D)=\omega_n r\), and \((\Omega^{\star})_D =B_r\). 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. MSC: 35P15 Estimates of eigenvalues in context of PDEs 35P30 Nonlinear eigenvalue problems and nonlinear spectral theory for PDEs Keywords:Zaremba problems; reverse Faber-Krahn inequality × Cite Format Result Cite Review PDF Full Text: DOI arXiv References: [1] T.V. Anoop and K. Ashok Kumar, On reverse Faber-Krahn inequalities, J. Math. Anal. Appl. 485 (2020), no. 1, 123766, 20. · Zbl 1437.35516 [2] T.V. Anoop, V. Bobkov and S. Sasi, On the strict monotonicity of the first eigenvalue of the \(p\)-Laplacian on annuli, Trans. Amer. Math. Soc. 370 (2018), no. 10, 7181-7199. · Zbl 1405.35064 [3] V. Bobkov and S. Kolonitskii, On qualitative properties of solutions for elliptic problems with the \(p\)-Laplacian through domain perturbations, Comm. 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