The Clamped Plate in Gauss Space
Annali di Matematica Pura ed Applicata
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 C_V 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 C_V ≥ 0.91 by solving an associated minimization problem in terms of parabolic cylinder functions.
Langford, Jeffrey J. and Chasman, Laura Mercredi. "The Clamped Plate in Gauss Space." Annali di Matematica Pura ed Applicata 195, no. 6 (2016) : 1977-2005.