On the Effectless Cut Method for Laplacian Eigenvalues in any dimensions

TL;DR

This paper investigates how to optimize the first Laplacian eigenvalue in axisymmetric doubly connected domains with Robin boundary conditions, proving spherical shells are optimal under certain constraints.

Contribution

It extends the effectless cut technique to higher dimensions and applies it to prove spherical shells maximize the eigenvalue under geometric constraints.

Findings

Spherical shells maximize the first Laplacian eigenvalue under given constraints.

Extension of the effectless cut technique to higher dimensions.

Combines isoperimetric inequalities with new geometric analysis.

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.