Abstract

Optimization of eigenvalues for differential operators on exterior domains is a topic of recent interest, where new challenges are connected with the presence of a non-empty essential spectrum. Most of the previous results are concerned with the Euclidean case. In this talk, we will discuss new results in the setting of the hyperbolic plane. We consider the Robin Laplacian in the exterior of a bounded simply-connected Lipschitz domain in the hyperbolic plane. The essential spectrum of this operator is [1/4, ∞). We show that, under convexity assumption on the domain, there exist discrete eigenvalues below 1/4 if, and only if, the Robin parameter is below a non-positive critical constant, which depends on the shape of the domain. We prove that the lowest Robin eigenvalue for the exterior of a bounded geodesically convex domain Ω 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 Ω. 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. The optimality of the exterior of the geodesic disk in the hyperbolic case was by far not evident in view of the new challenges such as the essential spectrum starting not from zero and non-criticality of the Neumann Laplacian in the exterior of a bounded convex domain. This talk is based on a joint work with Antonio Celentano and David Krejčiřík.