Another remark on constrained von Karman theories

TL;DR

This paper investigates a non-flat variant of the constrained von Karman theory for thin elastic films, demonstrating the existence of infinitely many stationary points for certain radially symmetric displacements.

Contribution

It proves that all admissible radially symmetric out-of-plane displacements are stationary points, and constructs examples with infinitely many stationary points, revealing complex solution structures.

Findings

Every admissible radially symmetric displacement is a stationary point.

Existence of data leading to infinitely many stationary points.

Theoretical insight into the solution landscape of constrained von Karman functionals.

Abstract

We study the natural non-flat version of the so-called "constrained von Karman" theory for thin nonlinearly elastic films. We prove that every (admissible) radially symmetric out-of-plane displacement on the unit disk is a stationary point. This allows us to construct data leading to constrained von Karman functionals which have infinitely many stationary points.