Bounds for the first eigenvalue of the elastically supported membrane on convex domains
This paper deals with lower bounds for the first eigenvalue of the following eigenvalue problem \[ \Delta u + \lambda u = 0 \;\text{in} \;\Omega, \qquad \frac{\partial u}{\partial n} + h u = 0 \;\text{on} \;\partial \Omega, \] where \(h > 0 \) is a constant and \(\Omega\) is a plane domain. The method is based on Barta's principle and is closely related to \textit{L. E. Payne} and \textit{G. A. Philippin} [SIAM J. Math. Anal. 14, 1154--1162 (1983; Zbl 0521.73056)] and \textit{R. P. Sperb} [Z. Angew. Math. Phys. 44, 639--653 (1993; Zbl 0785.35044)]. The solution of the torsion problem is extensively used as an auxiliary function. The paper contains numerical results and a nice example showing that the eigenvalue is not a monotone function of the domain.
scientific article; zbMATH DE number 2021120