2025/05/30 by Ji, Meng
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2505.24410
In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Ampère operator: \begincases amp;u≥φ in Ω amp;L wu=\tr( W D2u)≤ 0 in Ω amp;L wu= 0 in \ugt;φ\ amp;u=0 on ∂Ω, \endcases where W=(det D2 w) D2 w-1 is the matrix of cofactor of D2 w, w satisfies λ≤ det D2 w ≤ Λ and w=0 on ∂ Ω, φ is the obstacle with at least C2(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 C1,γ regularity for any γ∈ (0,1), provided that it is a strong solution in W2,nloc(Ω).