C^1, α 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: align* cases &uφ in Ω &L ₖu= (W D^2u) 0 in Ω &L ₖu= 0 in \u>φ\ &u=0 on Ω, cases align* where W= (D^2 w) D^2 w^-1 is the matrix of cofactor of D^2 w, w satisfies λ D^2 w Λ and w=0 on Ω, φ is the obstacle with at least C^2 (Ω) smoothness, Ω 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, γ regularity for any γ (0, 1), provided that it is a strong solution in W^2, n₋₎₂ (Ω).