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 flux
φ 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
φ =
π(
m2 + 2) and continue as
φ → ∞ whereas those with
m = 1 or 2 bifurcate subcritically and are terminated at
φ = 3
π and 4
π, respectively. The second variation can be evaluated directly for the circular state which is proven to be the minimizing solution only up to
φ = 3
π.
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