Minimization of the buckling load of a clamped plate with perimeter constraint

TL;DR

This paper investigates the shape optimization of plates under perimeter constraints to minimize buckling load, establishing convexity of minimizers in 2D and connectedness in higher dimensions.

Contribution

It proves convexity of minimizers in 2D and connectedness in higher dimensions for the buckling load problem with perimeter constraints.

Findings

Minimizers in 2D are convex sets.

In higher dimensions, minimizers are open and connected.

Existence of minimizers among convex sets with fixed perimeter.

Abstract

We look for minimizers of the buckling load problem with perimeter constraint in any dimension. In dimension 2, we show that the minimizing plates are convex; in higher dimension, by passing through a weaker formulation of the problem, we show that any optimal set is open and connected. For higher eigenvalues, we prove that minimizers exist among convex sets with prescribed perimeter.