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&L_{ w}u= 0 \\text{\\quad in } \\{u>\\varphi\\}\n  &u=0 \\text{\\quad on } \\partial\\Omega,\n  \\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 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regularity of the solution for the obstacle problem for the linearized Monge-Amp\\`ere operator","license":"http://arxiv.org/licenses/nonexclusive-distrib/1.0/","headline":"","cross_cats":[],"primary_cat":"math.AP","authors_text":"Meng Ji","submitted_at":"2025-05-30T09:49:35Z","abstract_excerpt":"In this paper, we study the regularity of the solution for the obstacle problem associated with the linearized Monge-Amp\\`ere operator: \\begin{align*}\n  \\begin{cases}\n  &u\\geq\\varphi \\text{\\quad in } \\Omega\n  &L_{ w}u=\\tr( W D^{2}u)\\leq 0 \\text{\\quad in } \\Omega\n  &L_{ w}u= 0 \\text{\\quad in } \\{u>\\varphi\\}\n  &u=0 \\text{\\quad on } \\partial\\Omega,\n  \\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 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