On the Effectless Cut Method for Laplacian Eigenvalues in any dimensions
Spectral Theory
2026-04-02 v1 Analysis of PDEs
Abstract
In this paper, we study the optimization of the first Laplacian eigenvalue on axisymmetric doubly connected domains under positive Robin boundary conditions. Under additional geometric constraints, we prove that spherical shells maximize this eigenvalue. Our approach combines known isoperimetric inequalities for mixed Laplacian eigenvalues with a higher-dimensional extension of the effectless cut technique introduced by Hersch to study multiply connected membranes of given area fixed along their boundaries.
Cite
@article{arxiv.2604.00976,
title = {On the Effectless Cut Method for Laplacian Eigenvalues in any dimensions},
author = {Vincenzo Amato and Nunzia Gavitone and Francesca de Giovanni},
journal= {arXiv preprint arXiv:2604.00976},
year = {2026}
}