The clamped plate in Gauss space

L. M. Chasman; Jeffrey J. Langford

doi:10.1007/s10231-016-0550-2

The clamped plate in Gauss space

L. M. Chasman, Jeffrey J. Langford

Mathematics (Morris)

Research output: Contribution to journal › Article › peer-review

3 Scopus citations

Abstract

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 C_V 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.

Original language	English (US)
Pages (from-to)	1977-2005
Number of pages	29
Journal	Annali di Matematica Pura ed Applicata
Volume	195
Issue number	6
DOIs	https://doi.org/10.1007/s10231-016-0550-2
State	Published - Dec 1 2016

Bibliographical note

Funding 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.

Publisher Copyright:
© 2016, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.

Keywords

Clamped plate
Comparison results
Gauss space
Parabolic cylinder functions
Symmetrization

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10.1007/s10231-016-0550-2

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title = "The clamped plate in Gauss space",

abstract = "In this paper, we study the analogue in Gauss space of Lord Rayleigh{\textquoteright}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{\textquoteright}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.",

keywords = "Clamped plate, Comparison results, Gauss space, Parabolic cylinder functions, Symmetrization",

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AU - Chasman, L. M.

AU - Langford, Jeffrey J.

N1 - Funding 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. Publisher Copyright: © 2016, Fondazione Annali di Matematica Pura ed Applicata and Springer-Verlag Berlin Heidelberg.

PY - 2016/12/1

Y1 - 2016/12/1

N2 - 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.

AB - 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.

KW - Clamped plate

KW - Comparison results

KW - Gauss space

KW - Parabolic cylinder functions

KW - Symmetrization

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SP - 1977

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JO - Annali di Matematica Pura ed Applicata

JF - Annali di Matematica Pura ed Applicata

IS - 6

ER -

The clamped plate in Gauss space

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