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The degenerate Monge-Ampère equations with the Neumann condition

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This work was supported by the National Natural Science Foundation of China (No. 11771214, No. 12001276).

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  • In this paper, we study a priori derivative estimates (up to the second order) of solutions for the Monge-Ampère equation $ \det D^{2}u = f(x) $ with the Neumann boundary value condition, which are independent of $ \inf f $. Based on these uniform estimates, the existence and uniqueness of the global $ C^{1,1} $ solution to the Neumann problem of the degenerate Monge-Ampère equation are established under the assumption $ f^{\frac{1}{n-1}}\in C^{1,1}(\bar{\Omega}) $.

    Mathematics Subject Classification: Primary: 35J96, 35J25; Secondary: 35J70, 35B65.


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