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<p>We prove a fourth order analogue of the Saint-Venant inequality: the mean deflection of a clamped plate under uniform transverse load is maximal for the ball, among plates of prescribed volume in any dimension of space. The method works in the Euclidean space, the hyperbolic space, and the sphere. Similar results for clamped plates under small compression and for the compliance under non-uniform loads are proved to hold in two dimensional Euclidean space, with the higher dimensional and curved cases of those problems left open.</p>

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Fourth Order Saint-Venant Inequalities: Maximizing Compliance and Mean Deflection Among Clamped Plates

  • 期刊论文

  • 出版模式：

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  • 发表日期：

    2025年9月1日

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  • Mark S. Ashbaugh,
  • Dorin Bucur,
  • Richard S. Laugesen,
  • Roméo Leylekian

摘要

We prove a fourth order analogue of the Saint-Venant inequality: the mean deflection of a clamped plate under uniform transverse load is maximal for the ball, among plates of prescribed volume in any dimension of space. The method works in the Euclidean space, the hyperbolic space, and the sphere. Similar results for clamped plates under small compression and for the compliance under non-uniform loads are proved to hold in two dimensional Euclidean space, with the higher dimensional and curved cases of those problems left open.