On the optimal strategy for an isotropic blocking problem

This article is concerned with a dynamic blocking problem, originally motivated by the control of wild fires. It is assumed that the region R(t) ⊂ ℝ ² burned by the fire is initially a disc, and expands with unit speed in all directions. To block the fire, a barrier Γ can be constructed in real time, so that the portion of the barrier constructed within time t has length ≤ σt, for some constant σ > 2. We prove that, among all barriers consisting of a single closed curve, the one which minimizes the total burned area is axisymmetric, and consists of an arc of circumference and two arcs of logarithmic spirals.