A Free Boundary Problem Related to Thermal Insulation: Flat Implies Smooth

We study the regularity of the interface for a new free boundary problem introduced by Caffarelli and Kriventsov. We show that for minimizers of the functional $$F_1(A,u) = \int_A |\nabla u|^2 d\mathcal{L}^n + \int_{\partial A} u^2 + \bar{C} \mathcal{L}^n(A)$$ over all pairs $(A,u)$ of open sets $A$ containing a fixed set $\Omega$ and functions $u\in H^1(A)$ which equal $1$ on $\Omega$, the boundary $\partial A$ locally coincides with the union of the graphs of two $C^{1,\alpha}$ functions near most points. Specifically, this happens at all points where the interface is trapped between two planes which are sufficiently close together. The proof combines ideas introduced by Ambrosio, Fusco, and Pallara for the Mumford-Shah functional with new arguments specific to the problem considered.