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 of degenerate fully nonlinear operators. More specifically, we consider viscosity solutions of

$ \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*} $

with $ \gamma>0 $, $ \phi \in C^{1, \alpha}(B_1) $ for some $ \alpha\in(0,1] $, a continuous boundary datum $ g $ and $ f\in L^\infty(B_1)\cap C^0(B_1) $ and prove that they are $ C^{1,\beta}(B_{1/2}) $ (and in particular at free boundary points) where $ \beta = \min\left\{\alpha, \frac{1}{\gamma+1}\right\} $. 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 $ \partial\{u>\phi\} $ has Hausdorff dimension less than $ n $ (and in particular zero Lebesgue measure). Our results are new even for degenerate problems such as

$ |Du|^\gamma \Delta u = \chi_{\{u>\phi\}} \quad \text{with}\quad \gamma>0. $