Existence of an optimal domain for the buckling load of a clamped plate with prescribed volume

TL;DR

This paper investigates the optimal shape of a clamped plate to maximize buckling load under volume constraints, using free boundary problems and penalization methods to identify the optimal domain.

Contribution

It introduces a novel free boundary formulation for the buckling problem with volume constraints and analyzes the effects of different penalization strategies.

Findings

Optimal domain existence established as penalization parameter tends to zero

Comparison of two penalization approaches for volume constraint enforcement

Provides mathematical foundation for shape optimization in buckling problems

Abstract

We formulate the minimization of the buckling load of a clamped plate as a free boundary value problem with a penalization term for the volume constraint. As the penalization parameter becomes small we show that the optimal shape problem with prescribed volume is solved. In addition, we discuss two different choices for the penalization term.