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We study the regularity of the interface for a new free boundary problemintroduced by Caffarelli and Kriventsov. We show that for minimizers of thefunctional \[  F_1(A,u) = \int_A |\nabla u|^2 d\mathcal{L}^n + \int_{\partial A} u^2 +\bar{C} \mathcal{L}^n(A) \] over all pairs $(A,u)$ of open sets $A$ containinga 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 allpoints where the interface is trapped between two planes which are sufficientlyclose together. The proof combines ideas introduced by Ambrosio, Fusco, andPallara for the Mumford-Shah functional with new arguments specific to theproblem considered.

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