Sharp regularity for degenerate obstacle type problems: A geometric approach

Abstract

We prove sharp regularity estimates for solutions of obstacle type problems driven by a class of degenerate fully nonlinear operators. More specifically, we consider viscosity solutions of \begin{document}$ \begin{equation*} \left\{ \begin{array}{rcll} |D u|^\gamma F(x, D^2u)& = & f(x)\chi_{\{u>\phi\}} & \rm{ in } B_1 u(x) & \geq & \phi(x) & \rm{ in } B_1 u(x) & = & g(x) & \rm{on } \partial B_1, \end{array} \right. \end{equation*} $\end{document} with \begin{document}$ \gamma>0 $\end{document} , \begin{document}$ \phi \in C^{1, \alpha}(B_1) $\end{document} for some \begin{document}$ \alpha\in(0,1] $\end{document} , a continuous boundary datum \begin{document}$ g $\end{document} and \begin{document}$ f\in L^\infty(B_1)\cap C^0(B_1) $\end{document} and prove that they are \begin{document}$ C^{1,\beta}(B_{1/2}) $\end{document} (and in particular at free boundary points) where \begin{document}$ \beta = \min\left\{\alpha, \frac{1}{\gamma+1}\right\} $\end{document} . Moreover, we achieve such a feature by using a recently developed geometric approach which is a novelty for these types of free boundary problems. Furthermore, under a natural non-degeneracy assumption on the obstacle, we prove that the free boundary \begin{document}$ \partial\{u>\phi\} $\end{document} has Hausdorff dimension less than \begin{document}$ n $\end{document} (and in particular zero Lebesgue measure). Our results are new even for degenerate problems such as \begin{document}$ |Du|^\gamma \Delta u = \chi_{\{u>\phi\}} \quad \text{with}\quad \gamma>0. $\end{document}