In this paper, we study the analogue in Gauss space of Lord Rayleigh’s conjecture for the clamped plate. We show that the first eigenvalue of the bi-Hermite operator in a bounded domain is bounded below by a constant CV times the corresponding eigenvalue of a half-space with the same Gaussian measure V. Similar results are established on unbounded domains. We use rearrangement methods similar to Talenti’s for the Euclidean clamped plate. We obtain our constant CV following the Euclidean approach of Ashbaugh and Benguria, and we find a numerical bound CV≥ 0.91 by solving an associated minimization problem in terms of parabolic cylinder functions.
Bibliographical noteFunding Information:
The authors would like to thank CIRM Luminy for support during the conference “Shape Optimization Problems and Spectral Theory,” where conversations surrounding this problem first began. The authors would also like to thank Richard Laugesen for posing this problem and for useful guidance and discussions. This research was partially supported by the University of Minnesota’s Faculty Development Single Semester Leave.
© 2016, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.
- Clamped plate
- Comparison results
- Gauss space
- Parabolic cylinder functions