2025/09/22 by T. V. Anoop, Anoop, T. V., В. И. Бобков +3
Mathematics · #26E05 #35P05 #35P15 #Analysis of PDEs (math.AP) #Analytic and geometric function theory #FOS: Mathematics #Holomorphic and Operator Theory #Point processes and geometric inequalities #Spectral Theory (math.SP)
paper · pdf · doi:10.48550/arxiv.2509.17480
openalex publication_date 2025/09/22 · openalex created_date 2025/10/16 · openalex updated_date 2026/07/28
We prove that among all doubly connected and elastically supported planar membranes Ω with prescribed values of the area |Ω| and the lengths of the inner and outer boundaries |∂ Ω_\rmin|1, |∂ Ω_\rmout|1 satisfying |∂ Ω_\rmout|12 - |∂ Ω_\rmin|12 = 4π|Ω|, the concentric annular membrane has the maximal fundamental frequency. The elastic constants h_\rmin, h_\rmout on ∂ Ω_\rmin, ∂ Ω_\rmout, respectively, are assumed to satisfy h_\rmin ⋅ h_\rmout ≥ 0 and can admit negative values and +∞, 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_\rmin ⋅ h_\rmout = 0 is proved using the method of interior parallels à la Payne & Weinberger, and it requires less restrictive assumptions on Ω. For the case h_\rmin ⋅ h_\rmout > 0, we develop the construction of the so-called ``effectless cut'' of Ω 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.