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 $$\label{E} E(u,Ω) = \int_Ω|\nabla u|^2 dX + \mathcal{H}^n(\{u>0\} \cap \{x_{n+1} = 0\}), \quad Ω\subset \R^{n+1},$$ among all functions $u\ge 0$ which are fixed on $\p Ω$. 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 $C^{2,α}$ surface except on a small singular set of Hausdorff dimension $n-3$. We also obtain $C^{2,α}$ regularity of Lipschitz free boundaries of viscosity solutions associated to this problem.

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