Optimisation of the lowest Robin eigenvalue in exterior domains of the hyperbolic plane

TL;DR

This paper investigates the lowest Robin eigenvalue in the exterior of convex domains in the hyperbolic plane, establishing bounds and optimality conditions based on domain shape and geometric constraints.

Contribution

It proves that the geodesic disk maximizes the lowest Robin eigenvalue among convex exterior domains with fixed area or perimeter, and characterizes the critical Robin parameter.

Findings

The essential spectrum of the Robin Laplacian starts at 1/4.

The lowest eigenvalue is maximized by the geodesic disk among convex domains.

A reverse inequality for critical constants is established.

Abstract

We consider the Robin Laplacian in the exterior of a bounded simply-connected Lipschitz domain in the hyperbolic plane. We show that the essential spectrum of this operator is $[\frac14,\infty)$ and that, under convexity assumption on the domain, there exist discrete eigenvalues below $\frac14$ if, and only if, the Robin parameter is below a non-positive critical constant, which depends on the shape of the domain. As the main result, we prove that the lowest Robin eigenvalue for the exterior of a bounded geodesically convex domain $\Omega$ in the hyperbolic plane does not exceed such an eigenvalue for the exterior of the geodesic disk, whose geodesic curvature of the boundary is not smaller than the averaged geodesic curvature of the boundary of $\Omega$. This result implies as a consequence that under fixed area or fixed perimeter constraints the exterior of the geodesic disk maximises the lowest Robin eigenvalue among exteriors of bounded geodesically convex domains. Moreover, we obtain under the same geometric constraints a reverse inequality between the critical constants.