Local Spectral Optimisation for Robin Problems with Negative Boundary Parameter on Quadrilaterals

TL;DR

This paper studies the Robin eigenvalue problem on quadrilaterals, proving the square's local maximality for the first eigenvalue with negative boundary parameters and exploring asymptotic optimality for extreme parameter values.

Contribution

It establishes the square as a local maximiser of the first Robin eigenvalue on quadrilaterals and provides asymptotic analysis for extreme Robin parameters.

Findings

The square is a local maximiser of the first Robin eigenvalue among quadrilaterals.

Asymptotic results relate to the optimality of the square for large or small Robin parameters.

The study advances understanding of eigenvalue optimization with negative boundary parameters.

Abstract

We investigate the Robin eigenvalue problem for the Laplacian with negative boundary parameter on quadrilateral domains of fixed area. In this paper, we prove that the square is a local maximiser of the first eigenvalue with respect to the Hausdorff metric. We also provide asymptotic results relating to the optimality of the square for extreme values of the Robin parameter.