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

2025/01/25 by Ashbaugh, Mark, Bucur, Dorin, Laugesen, Richard S. +1 · 1 citation
#35J35 #Analysis of PDEs (math.AP) #FOS: Mathematics

paper · doi:10.48550/arxiv.2501.15026

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.

Cited by

• Extremal problems for clamped plates under tension

Related

• Boundary regularity of a fourth order Alt-Caffarelli problem and applications to the minimization of the critical buckling load
• Extremal problems for clamped plates under tension
• Optimality conditions for the buckling of a clamped plate
• Asymptotic behaviour of solutions of fourth order Dirichlet problems
• Existence of an optimal domain for minimizing the fundamental tone of a clamped plate of prescribed volume in arbitrary dimension