Regularity of Lipschitz free boundaries for the thin one-phase problem (Q496449)

In this very nice paper, the authors study the free boundary for the thin one-phase problem. This problem consists of minimizing the energy functional \[ E(u,\Omega):=\int_{\Omega}|\nabla u|^2dX +\mathcal{H}^n(\{u>0\cap\{x_{n+1}=0\}), \quad \Omega\subset \mathbb{R}^{n+1}, \] among all functions \(u\geq 0\) which are fixed on \(\partial\Omega\). It is assumed that \(\Omega\) is symmetric with respect to \(\{x_{n+1}=0\}\) and \(u\) is even with respect to \(x_{n+1}\). The main focus of this paper is on the free boundary of minimizers \(u\), that is, the set \[ F(u):=\partial_{\mathbb{R}^n}\{u(x,0)>0\}\cap\Omega\subset \mathbb{R}^n. \] The first main result of the paper states that \(F(u)\) is locally a \(C^{2,\alpha}\) surface except on a singular set \(\Sigma_u\subset F(u)\) such that \(\mathcal{H}^s(\Sigma_u)=0\) for \(s>n-3\). Moreover, it is proved that \(F(u)\) has locally finite \(\mathcal{H}^{n-1}\) measure. As a consequence of this result, one obtains that, when \(n=2\), the free boundaries of minimizers are \(C^{2,\alpha}\). The second main result of this paper concerns the regularity of Lipschitz free boundaries of viscosity solutions to the Euler-Lagrange equation associated to the minimization problem for \(E\). More precisely, the authors consider the problem \[ \begin{aligned} \Delta u=0 \text{ in } \Omega\setminus \{(x,0) \;| \;u(x,0)=0\}, \\ \lim\limits_{t\rightarrow 0^+}\frac{u(x_0+t\nu(x_0),0)}{\sqrt{t}} =1, \\ x_0\in F(u):=\partial_{\mathbb{R}^n}\{u(x,0)>0\}\cap\Omega. \end{aligned} \] Here, \(\nu(x_0)\) is the normal to \(F(u)\) at \(x_0\) pointing toward \(\{x \;| \;u(x,0)>0\}\). The authors prove that if \(u\) is a viscosity solution of this problem in \(B_1\) with \(0\in F(u)\) and such that \(F(u)\) is a Lipschitz graph in the direction \(e_n\), then \(F(u)\cap B_{1/2}\) is a \(C^{2,\alpha}\) graph for any \(\alpha<1\). The main tools used in this paper are a Weiss-type monotonicity formula proved for minimizers of \(E\) and viscosity solutions which have Lipschitz free boundaries, and \(C^{2,\alpha}\) estimates for flat solutions. The Weiss-type monotonicity formula is used to perform a blow-up analysis near the free boundary and reduces the regularity problem to classifying the so-called global cones, that is, homogeneous global solutions of degree \(1/2\). The \(C^{2,\alpha}\) estimates for flat solutions lead to the conclusion that all Lipschitz cones are trivial, which, combined with a dimension reduction argument, concludes the proof of the first main result, while the flatness theorem and the monotonicity formula lead to the second main result.