A minimum problem with free boundary for a degenerate quasilinear operator (Q1776572)

The authors prove \(C^{1,\alpha}\) regularity (near flat points) of the free boundary \(\partial\{u>0\}\cap\Omega\) in the Alt-Caffarelli type minimum problem for the \(p\)-Laplace operator: \[ J(u) = \int_{\Omega} (|\nabla u|^{p} + \lambda^{p}\chi_{\{u>0\}}) \,dx \to \min \quad (1<p<\infty). \]