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Fourth order Saint-Venant inequalities: maximizing compliance and mean deflection among clamped plates

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

TL;DR

This work establishes a fourth-order Saint-Venant-type inequality for clamped plates under constant load: among plates of fixed volume, the mean deflection (and hence the compliance) is maximized by a geodesic ball in Euclidean, spherical, and hyperbolic spaces for zero compression, with explicit forms and exact constants. The authors develop a Talenti-style rearrangement framework and a two-ball auxiliary problem to prove that the ball minimizes the energy $E(\Omega,0)$, yielding equality only for balls (modulo $H^2$-capacity-zero sets). They also obtain a partial result in the planar case for small compression, extend explicit formulas to 2D curved geometries, and solve the nonconstant-load compliance problem in the plane, while highlighting open problems in higher dimensions. The paper integrates variational methods, elliptic rearrangements, and delicate transport-type arguments to generalize Saint-Venant-type optimality to higher-order elasticity problems on spaces of constant curvature. The results have implications for isoperimetric-type extremals in elasticity and provide a rigorous basis for predicting that ball-like domains maximize deflection-related quantities under prescribed volume and loads. Open problems include extending the total- and mean-deflection maximization to higher dimensions and curved spaces for variable loads.

Abstract

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 Euclidean space, 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.

Fourth order Saint-Venant inequalities: maximizing compliance and mean deflection among clamped plates

TL;DR

This work establishes a fourth-order Saint-Venant-type inequality for clamped plates under constant load: among plates of fixed volume, the mean deflection (and hence the compliance) is maximized by a geodesic ball in Euclidean, spherical, and hyperbolic spaces for zero compression, with explicit forms and exact constants. The authors develop a Talenti-style rearrangement framework and a two-ball auxiliary problem to prove that the ball minimizes the energy , yielding equality only for balls (modulo -capacity-zero sets). They also obtain a partial result in the planar case for small compression, extend explicit formulas to 2D curved geometries, and solve the nonconstant-load compliance problem in the plane, while highlighting open problems in higher dimensions. The paper integrates variational methods, elliptic rearrangements, and delicate transport-type arguments to generalize Saint-Venant-type optimality to higher-order elasticity problems on spaces of constant curvature. The results have implications for isoperimetric-type extremals in elasticity and provide a rigorous basis for predicting that ball-like domains maximize deflection-related quantities under prescribed volume and loads. Open problems include extending the total- and mean-deflection maximization to higher dimensions and curved spaces for variable loads.

Abstract

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 Euclidean space, 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.
Paper Structure (15 sections, 9 theorems, 146 equations)

This paper contains 15 sections, 9 theorems, 146 equations.

Table of Contents

  1. Introduction
  2. Constant load results ($\rho \equiv 1$): zero compression ($\sigma=0$) in Euclidean, spherical and hyperbolic spaces, and small compression in the plane
  3. Mean deflection and minimal energy
  4. Zero compression
  5. Extremal energies under zero compression
  6. Small positive compression
  7. Proof of Theorem \ref{['as10general']} --- constant load, zero compression
  8. Alternative (partial) proof of Theorem \ref{['as10general']} by explicit formulas
  9. Euclidean case ($M={\mathbb{R}^N}, \sigma=0$)
  10. $2$-dim spherical case ($M={\mathbb S}^2, \sigma=0$)
  11. $2$-dim hyperbolic case ($M={\mathbb H}^2, \sigma=0$)
  12. Proof of Theorem \ref{['as13']} --- positive compression in Euclidean plane
  13. The compliance problem with nonconstant load
  14. Possible alternative strategies
  15. Talenti-style elliptic rearrangement on constant curvature spaces

Key Result

Theorem 2.1

Let $\Omega \subset M$ be a nonempty open set with finite measure, and in the spherical case assume $1 \notin H^2_0(\Omega)$. If $\Omega^*$ is a geodesic ball with the same volume as $\Omega$ then the unique solution $u_\Omega$ of as01zero satisfies or equivalently Equality occurs if and only if $\Omega$ coincides with a geodesic ball, up to a set of $H^2$-capacity zero.

Theorems & Definitions (19)

  • Theorem 2.1: Ball minimizes the energy when compression is absent
  • Proposition 2.2: Maximal mean deflection under zero compression
  • Theorem 2.3: Disk minimizes the energy, under compression
  • Remark 2.4
  • Remark 2.5
  • Proposition 3.1
  • proof
  • Remark 3.2
  • Proposition 5.1
  • proof
  • ...and 9 more