Sharp regularity for degenerate obstacle type problems: A geometric approach

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