Free-boundary regularity for a problem arising in superconductivity
For any \(\rho >0\) and \(x_0\) given in \(\mathbb{R}^n\) consider the problem of finding \(u \in C^{1,1}(B_{\rho}(x_0))\) such that \[ \Delta u=\chi_{\{| \nabla u| >0\}} \] and \[ | \nabla u(x_0)| =0,\,\,\,| u(x)-u(x_0)| \leq c(1+| x| ^2) \] with \(c >0\). It is known that for all \((c,\rho, x)\) all solutions \(u\) have uniform \(C^{1,1}\) estimate depending only on \(c\) and \(n\), and that the free boundary \(\partial \{| \nabla u| >0\}\) has locally finite \((n-1)\)-Hausdorff measure. Let \(P(c,\rho,x_0)\) denote the class of all solutions of the problem above and for each of them consider the sets \[ \Omega = \text{int} {\{| \nabla u| >0}\},\,\,\,\,\Omega^c=\mathbb{R}^n\backslash \Omega. \] The density function of \(\Omega^c\) is defined by \[ \delta_{\rho}(u)=\dfrac{MD(\Omega^c\cap B_{\rho})}{\rho} \] where \(MD\) denotes the minimal diameter (i.e. the infimum of the thickness of layers containing the set). The main result of the paper is the following. There exists a modulus of continuity \(\sigma\) such that if \(\delta_{\rho_0}(u)>\sigma(\rho_0)\) for some \(\rho_0<\frac{1}{2}\), then for any \(u\) in the class \(P(c,1,x_0)\) the boundary \(\partial\{| \nabla u| >0\}\) is a \(C^1\) graph in \(B(x_0,c_0\rho^2_0)\), where \(c_0\) depends only on \(c\) and on \(n\). The main tool employed is the blow up technique.
scientific article; zbMATH DE number 2101871