Reverse Faber-Krahn inequalities for Zaremba problems (Q6654803)

Consider following problem and let \(\tau_{1, q}(\Omega)\) be its first eigenvalue, \N\[ \N\begin{aligned} -\Delta_p u & =\tau\left(\int_{\Omega}|u|^q d x\right)^{(p-q) / q}|u|^{q-2} u & & \text { in } \Omega, \\\Nu & =0 & & \text { on } \partial \Omega_D, \\\N\frac{\partial u}{\partial \eta} & =0 & & \text { on } \partial \Omega \backslash \partial \Omega_D, \end{aligned} \N\]\Nwhere \(\Omega\) is a domain in \(\mathbb{R}^n\) \((n \geq 2)\) of the form \(\Omega=\Omega_{\text {out }} \backslash \overline{\Omega_{\text {in }}}\), \(\Omega_D\) is either \(\Omega_{\text {out }}\) or \(\Omega_{\mathrm{in}}\), \(p \in(1, \infty)\), and \(q \in[1, p]\).\N\NAssuming that \(\Omega_D\) is convex, the authors establish the following reverse Faber-Krahn inequality\N\[\N\tau_{1, q}(\Omega) \leq \tau_{1, q}\left(\Omega^{\star}\right) \N\]\Nwhere \(\Omega^{\star}=B_R \backslash \overline{B_r}\) is a concentric annular region in \(\mathbb{R}^n\) having the same Lebesgue measure as \(\Omega\) and such that\N\N(i) (when \(\Omega_D=\Omega_{\text {out }}\)) \(W_1\left(\Omega_D\right)=\omega_n R^{n-1}\), and \(\left(\Omega^{\star}\right)_D=B_R\),\N\N(ii) (when \(\Omega_D=\Omega_{\text {in }}\)) \(W_{n-1}\left(\Omega_D\right)=\omega_n r\), and \(\left(\Omega^{\star}\right)_D=B_r\).\N\NHere \(W_i\left(\Omega_D\right)\) is the \(i^{\text {th }}\) quermassintegral of \(\Omega_D\).