A minimum problem with free boundary for a degenerate quasilinear operator

In this paper we prove \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$C^{1,\alpha}$\end{document} regularity (near flat points) of the free boundary \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$\partial\{u > 0\}\cap\Omega$\end{document} in the Alt-Caffarelli type minimum problem for the p-Laplace operator: \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$J(u)=\int_\Omega\left( |\nabla u|^p + \lambda^p\chi_{\{u>0\}}\right)dx\rightarrow \min\qquad (1<p<\infty).$$\end{document}