Bounds and extremal domains for Robin eigenvalues with negative boundary parameter

TL;DR

This paper derives new bounds for the first Robin eigenvalue with negative boundary parameter, analyzes extremal domains, and explores how eigenvalues change with boundary conditions and domain shape.

Contribution

It introduces novel bounds for Robin eigenvalues with negative parameters, including extremal domain characterizations and numerical optimization insights.

Findings

Disk maximizes the first eigenvalue at fixed perimeter for all negative boundary parameters.

Bounds for eigenvalues of balls and spherical shells are sharp and explicitly derived.

Bifurcation analysis reveals domain shape changes as boundary parameter becomes large and negative.

Abstract

We present some new bounds for the first Robin eigenvalue with a negative boundary parameter. These include the constant volume problem, where the bounds are based on the shrinking coordinate method, and a proof that in the fixed perimeter case the disk maximises the first eigenvalue for all values of the parameter. This is in contrast with what happens in the constant area problem, where the disk is the maximiser only for small values of the boundary parameter. We also present sharp upper and lower bounds for the first eigenvalue of the ball and spherical shells. These results are complemented by the numerical optimisation of the first four and two eigenvalues in 2 and 3 dimensions, respectively, and an evaluation of the quality of the upper bounds obtained. We also study the bifurcations from the ball as the boundary parameter becomes large (negative).