The obstacle problem for a class of degenerate fully nonlinear operators

We study the obstacle problem for fully nonlinear elliptic operators with an anisotropic degeneracy on gradient: $$ \left{\begin{array}{rll} \min\left{f-|Du|^\gamma F(D^2u),u-\phi\right} &= 0 & \textrm{ in } \Omega,\ u = g \partial \Omega, \end{array}\right. some parameter $\gamma\geq 0$, uniformly operator $F$, bounded source term $f$, and suitably smooth $\phi$ boundary datum $g$. obtain existence/uniqueness of solutions prove sharp regularity estimates at free points, namely $\partial{u>\phi} \cap \Omega$. In particular, homogeneous case ($f\equiv0$) we get that are $C^{1,1}$ sense they detach from a quadratic fashion, thus beating optimal allowed such degenerate operators. also several non-degeneracy properties partial results regarding boundary. These first problems driven by type non-divergence form novelty even simpler prototype given $\mathcal{G}\[u] |Du|^\gamma\Delta u$, $\gamma >0$ $f \equiv 1$.