An isoperimetric inequality for the perturbed Robin bi-Laplacian in a planar exterior domain

TL;DR

This paper establishes an isoperimetric inequality for the lowest eigenvalue of a perturbed Robin bi-Laplacian operator in a planar exterior domain, showing the disk maximizes the eigenvalue under certain curvature and eigenfunction conditions.

Contribution

It introduces a new spectral inequality for a perturbed Robin bi-Laplacian in exterior domains, extending classical isoperimetric results to this setting with a focus on the disk as extremizer.

Findings

The essential spectrum of the operator is the positive semi-axis.

Negative discrete spectrum exists if and only if the boundary parameter is negative.

The disk maximizes the lowest eigenvalue among convex exterior domains under curvature constraints.

Abstract

In the present paper we introduce the perturbed two-dimensional Robin bi-Laplacian in the exterior of a bounded simply-connected $C^2$-smooth open set. The considered perturbation is of lower order and corresponds to tension. We prove that the essential spectrum of this operator coincides with the positive semi-axis and that the negative discrete spectrum is non-empty if, and only if, the boundary parameter is negative. As the main result, we obtain an isoperimetric inequality for the lowest eigenvalue of such a perturbed Robin bi-Laplacian with a negative boundary parameter in the exterior of a bounded convex planar set under the constraint on the maximum of the curvature of the boundary with the maximizer being the exterior of the disk. The isoperimetric inequality is proved under the additional assumption that to the lowest eigenvalue for the exterior of the disk corresponds a radial eigenfunction. We provide a sufficient condition in terms of the tension parameter and the radius of the disk for this property to hold.