  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.
