Unstable free boundary problems in optimal control theory: existence and regularity

We establish the first general regularity result for constrained optimal control problems arising naturally in mathematical physics and mathematical biology. Namely, we prove that for a large class of problems of the form "maximise ∫ψ(Θm)−c∫m where −ΔΘm=m Θm+B(x,Θm), under the constraint 0≤m≤1 a.e.", the solution m∗ is bang-bang, in the sense that m∗=χE∗, and that ∂E∗ is smooth up to a (d−2)-dimensional subset. Moreover, we prove that the solutions to the volume constrained problem "maximise ∫ψ(Θm) where −ΔΘm=m Θm+B(x,Θm), under the constraint 0≤m≤1 a.e and ∫m=m0" are bang-bang in the sense that m∗=χE∗ and that, in the two-dimensional case, ∂E∗ is a finite union of smooth curves. This is done via reduction to an unstable free boundary problem, the regularity analysis of which was pioneered by Monneau & Weiss and Chanillo, Kenig & To. In our case, the free boundary is not minimising, and the laplacian of the state function is sign-changing, which creates significant difficulties, in particular regarding the non-degeneracy of blow-ups. This requires a new approach blending tools from optimal control theory, free boundary and measure theory to establish the regularity of the free boundary.