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

TL;DR

This paper establishes a fourth order Saint-Venant inequality showing that among all plates of a given volume, the ball maximizes mean deflection under uniform load, extending classical results to various geometries.

Contribution

It introduces a new inequality for clamped plates, proving maximal mean deflection for the ball in multiple geometries, and extends classical second-order results to fourth order.

Findings

The mean deflection is maximized by the ball in Euclidean, hyperbolic, and spherical spaces.

The method applies to uniform loads and is extendable to certain non-uniform load problems.

Results for small compression and compliance in higher dimensions remain open.

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.