$C^{1,\alpha}$ regularity of the solution for the obstacle problem for the linearized Monge-Amp\`ere operator

In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Amp\`ere operator: \begin{align*} \begin{cases} &u\geq\varphi \text{\quad in } \Omega &L_{ w}u=\tr( W D^{2}u)\leq 0 \text{\quad in } \Omega &L_{ w}u= 0 \text{\quad in } \{u>\varphi\} &u=0 \text{\quad on } \partial\Omega, \end{cases} \end{align*} where $W=(\det D^{2} w) D^{2} w^{-1}$ is the matrix of cofactor of $D^{2} w$, $w$ satisfies $\lambda \leq \det D^{2} w \leq \Lambda$ and $w=0$ on $\partial \Omega$, $\varphi$ is the obstacle with at least $C^{2}(\bar{\Omega})$ smoothness, $\Omega$ is an open bounded convex domain. We show the existence and uniqueness of a viscosity solution by using Perron's method and the comparison principle. Our primary result is to prove that the solution exhibits local $C^{1,\gamma}$ regularity for any $\gamma \in (0,1)$, provided that it is a strong solution in $W^{2,n}_{\text{loc}}(\Omega)$.