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

In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Ampère operator: \begin{align*} \begin{cases} &u\geq\varphi $\text{in }$ Ω &L_{ w}u=\tr( W D^{2}u)\leq 0 $\text{in }$ Ω &L_{ w}u= 0 $\text{in }$ {u>\varphi} &u=0 $\text{on }$ \partialΩ, \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 $\lambda \le \det D^{2}w\le \mathrm{\Lambda }$ and $w=0$ on $\mathrm{\partial }\mathrm{\Omega }$, $\phi$ is the obstacle with at least $C^2(\bar{\mathrm{\Omega }})$ smoothness, $\mathrm{\Omega }$ 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 $W_{\text{loc}}^{2,n}(\mathrm{\Omega })$.