Convexity and uniqueness in a free boundary problem arising in combustion theory.
We consider solutions to a free boundary problem for the heat equation, describing the propagation of flames. Suppose there is a bounded domain Ω ⊂ QT = Rn x (0,T) for some T > 0 and a function u > 0 in Ω such that ut = Δu, in Ω, u = 0 and |∇u| = 1, on Γ := ∂Ω ∩ QT, u(·,0) = u0, on Ω0, ...