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

Abstract

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\geqφ\text{\quad in } Ω &L_{ w}u=\tr( W D^{2}u)\leq 0 \text{\quad in } Ω &L_{ w}u= 0 \text{\quad in } \{u>φ\} &u=0 \text{\quad 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 $\hat{I}»\le \det D^{2}w\le \hat{I}›$ and $w=0$ on $\partial \hat{I}\text{c◯}$, $\ddot{I}†$ is the obstacle with at least $C^2(\bar{\hat{I}}\text{c◯})$ smoothness, $\hat{I}\text{c◯}$ 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 $C^{1,{\hat{I}}^3}$ regularity for any ${\hat{I}}^3\in (0,1)$, provided that it is a strong solution in $W_{\text{loc}}^{2,n}(\hat{I}\text{c◯})$.