Symmetry Breaking Bifurcation of Membranes with Boundary

TL;DR

This paper demonstrates a symmetry breaking bifurcation in membrane solutions with fixed boundary using bifurcation theory, simplifying the problem via a second order reduction.

Contribution

It applies Crandall and Rabinowitz's bifurcation theory to a membrane problem, revealing symmetry breaking phenomena with a novel reduction approach.

Findings

Existence of symmetry breaking bifurcation in membrane solutions

Application of bifurcation theory to membrane equations

Reduction of fourth order problem to second order

Abstract

We use a bifurcation theory due to Crandall and Rabinowitz to show the existence of a symmetry breaking bifurcation of a specific one parameter family of axially symmetric disc type solutions of a membrane equation with fixed boundary. In place of working directly with the fourth order membrane equation, it is replaced by a second order reduction found in [16].