Reverse isoperimetric inequality for the lowest Robin eigenvalue of a triangle

TL;DR

This paper investigates the lowest Robin eigenvalue of a triangle, proving local maximality of the equilateral triangle under certain boundary conditions and exploring conditions for global optimality.

Contribution

It establishes the local maximality of the equilateral triangle for the lowest Robin eigenvalue under small negative boundary parameters and provides conditions for its global optimality.

Findings

Equilateral triangle is a local maximiser for small negative boundary parameters.

Sufficient conditions for global optimality of the equilateral triangle are derived.

Discussion of fixed perimeter constraint impacts on eigenvalue optimization.

Abstract

We consider the Laplace operator on a triangle, subject to attractive Robin boundary conditions. We prove that the equilateral triangle is a local maximiser of the lowest eigenvalue among all triangles of a given area provided that the negative boundary parameter is sufficiently small in absolute value, with the smallness depending on the area only. Moreover, using various trial functions, we obtain sufficient conditions for the global optimality of the equilateral triangle under fixed area constraint in the regimes of small and large couplings. We also discuss the constraint of fixed perimeter.