A constrained variational model for radial symmetry breaking

Abstract

A simple constrained minimization problem with an integral constraint describes a symmetry breaking of a circular front around a point source. As a single control parameter, the total ux from the source, is varied, apparently polygonal solutions with an arbitrary number of corners m are shown to bifurcate from the circular solution. Our asymptotic analysis shows that the branches with m 3 bifurcate supercritically at = (m 2 + 2) and continue as ! 1 whereas those with m = 1 or 2 bifurcate subcritically and are terminated at = 3 and 4 , respectively. The second variation can be evaluated directly for the circular state which is proven to be the minimizing solution only up to = 3 .