Regularity of Lipschitz free boundaries for the thin one-phase problem

We study regularity properties of the free boundary for the thin one-phase problem which consists of minimizing the energy functional $$\label{E} E(u,\Omega) = \int_\Omega |\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad \Omega \subset \R^{n+1},$$ among all functions $u\ge 0$ which are fixed on $\p \Omega$. We prove that the free boundary $F(u)=\p_{\R^n}\{u>0\}$ of a minimizer $u$ has locally finite $\mathcal{H}^{n-1}$ measure and is a $C^{2,\alpha}$ surface except on a small singular set of Hausdorff dimension $n-3$. We also obtain $C^{2,\alpha}$ regularity of Lipschitz free boundaries of viscosity solutions associated to this problem.