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SUMMARY:Optimization of the lowest Robin eigenvalue in exterior domains of
  the hyperbolic plane - Vladimir Lotoreichik (Academy of Sciences of the C
 zech Republic)
DTSTART:20260311T140000Z
DTEND:20260311T150000Z
UID:TALK242077@talks.cam.ac.uk
DESCRIPTION:Optimization of eigenvalues for differential operators on exte
 rior domains is a topic of recent interest\, where new challenges are conn
 ected with the presence of a non-empty essential spectrum. Most of the pre
 vious results are concerned with the Euclidean case. In this talk\, we wil
 l discuss new results in the setting of the hyperbolic plane.\nWe consider
  the Robin Laplacian in the exterior of a bounded simply-connected Lipschi
 tz domain in the hyperbolic plane. The essential spectrum of this operator
  is [1/4\, &infin\;). We show that\, under convexity assumption on the dom
 ain\, there exist discrete eigenvalues below 1/4 if\, and only if\, the Ro
 bin parameter is below a non-positive critical constant\, which depends on
  the shape of the domain. We prove that the lowest Robin eigenvalue for th
 e exterior of a bounded geodesically convex domain &Omega\; in the hyperbo
 lic plane does not exceed such an eigenvalue for the exterior of the geode
 sic disk\, whose geodesic curvature of the boundary is not smaller than th
 e averaged geodesic curvature of the boundary of &Omega\;. This result imp
 lies as a consequence that under fixed area or fixed perimeter constraints
  the exterior of the geodesic disk maximises the lowest Robin eigenvalue a
 mong exteriors of bounded geodesically convex domains. Moreover\, we obtai
 n under the same geometric constraints a reverse inequality between the cr
 itical constants. The optimality of the exterior of the geodesic disk in t
 he hyperbolic case was by far not evident in view of the new challenges su
 ch as the essential spectrum starting not from zero and non-criticality of
  the Neumann Laplacian in the exterior of a bounded convex domain.\nThis t
 alk is based on a joint work with Antonio Celentano and David Krejčiř&ia
 cute\;k.
LOCATION:Seminar Room 2\, Newton Institute
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