Optimisation and monotonicity of the second Robin eigenvalue on a planar exterior domain

TL;DR

This paper investigates the second Robin eigenvalue for the Laplace operator in exterior planar domains, proposing conjectures, proving special cases, and establishing monotonicity results under geometric constraints.

Contribution

It introduces a conjecture about maximizing the second eigenvalue by a disk, proves it for convex domains, and shows monotonicity for star-shaped, symmetric sets.

Findings

Second eigenvalue maximized by the disk under certain constraints.

Proved the conjecture for convex exterior domains.

Established monotonicity for strictly star-shaped, symmetric domains.

Abstract

We consider the Laplace operator in the exterior of a compact set in the plane, subject to Robin boundary conditions. If the boundary coupling is sufficiently negative, there are at least two discrete eigenvalues below the essential spectrum. We state a general conjecture that the second eigenvalue is maximised by the exterior of a disk under isochoric or isoperimetric constraints. We prove an isoelastic version of the conjecture for the exterior of convex domains. Finally, we establish a monotonicity result for the second eigenvalue under the condition that the compact set is strictly star-shaped and centrally symmetric.