Gauss curvature flow with an obstacle

We consider the obstacle problem for the Gauss curvature flow with an exponent α\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\alpha $$\end{document}. Under the assumption that both the obstacle and the initial hypersurface are strictly convex closed hypersurfaces and that the obstacle is enclosed by the initial hypersurface, uniform estimates are obtained for several curvatures via a penalty method. We also prove that when the hypersurface is two dimensional with 0<α≤1\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$0<\alpha \le 1$$\end{document}, the solution of the Gauss curvature flow with an obstacle exists for all time with bounded principal curvatures {λi}\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\{\lambda _i\}$$\end{document}, where the upper bound is uniform, and the lower bound depends on the distance from the free boundary. Moreover, we show that there exists a finite time T∗\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$T_*$$\end{document} after which the solution becomes stationary in the same shape as the obstacle.