Reverse Faber-Krahn inequality for planar doubly connected domains

We prove that among all doubly connected and elastically supported planar membranes $\Omega$ with prescribed values of the area $|\Omega|$ and the lengths of the inner and outer boundaries $|\partial \Omega_{\rm{in}}|_1$, $|\partial \Omega_{\rm{out}}|_1$ satisfying $|\partial \Omega_{\rm{out}}|_1^2 - |\partial \Omega_{\rm{in}}|_1^2 = 4\pi |\Omega|$, the concentric annular membrane has the maximal fundamental frequency. The elastic constants $h_{\rm{in}}$, $h_{\rm{out}}$ on $\partial \Omega_{\rm{in}}$, $\partial \Omega_{\rm{out}}$, respectively, are assumed to satisfy $h_{\rm{in}} \cdot h_{\rm{out}} \geq 0$ and can admit negative values and $+\infty$, the latter being understood as a fixation of the membrane on the corresponding part of the boundary. Our study extends and unifies several existing results in the literature. The case $h_{\rm{in}} \cdot h_{\rm{out}} = 0$ is proved using the method of interior parallels \`a la Payne & Weinberger, and it requires less restrictive assumptions on $\Omega$. For the case $h_{\rm{in}} \cdot h_{\rm{out}} > 0$, we develop the construction of the so-called ``effectless cut'' of $\Omega$ described in terms of the gradient flow of the first eigenfunction. This concept was originally introduced by Weinberger and used by Hersch in the fixed boundary case, whose arguments we also revise.