On the curvature of the free boundary for the obstacle problem in two dimensions (Q1885253)

Let \(B\) be a ball in \(\mathbb{R}^2\) and consider the obstacle problem: \[ u\in C^1(B),\,\,\,u\geq 0,\,\,\,\Delta u=\chi_{\Omega}\,\,\,{\text{in}}\,\,\, B \] where \(\Omega\) is the noncoincidence set and \(\chi_{\Omega}\) is its characteristic function. The condition \(u\in C^1 (B)\) implies \(u=| \nabla u| =0\) on the free boundary \(\partial \Omega \backslash B\). Fixed boundary conditions are prescribed away from \(B\). The aim of the paper is to prove that if \(B\) is centered at the origin and we take the coordinate \(x, y\), the following result holds true: if \(\Omega^+ =\Omega \cap \{y>0\}\) is relatively compact in \(B\), then there exists a function \(r(x)\) such that \[ \Omega^+ =U_{x\in\Omega\cap\mathbb{R}}B(x,r(x))^+, \] where \(B^+(x,r)\) is the half disc with center at the point \((x,0)\) and of radius \(r\), lying in the half plane \(\{y>0\}\).