The degenerate Monge-Ampère equations with the Neumann condition

Juhua Shi; Feida Jiang

doi:10.3934/cpaa.2020297

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2021, Volume 20, Issue 2: 915-931. Doi: 10.3934/cpaa.2020297

This issue Previous Article Inequalities of Hermite-Hadamard type for higher order convex functions, revisited Next Article Ground states for a class of quasilinear Schrödinger equations with vanishing potentials

The degenerate Monge-Ampère equations with the Neumann condition

Juhua Shi^1, and
Feida Jiang^2, ,

1.
School of Science, Nanjing University of Science and Technology, Nanjing 210094, China
2.
School of Mathematics and Shing-Tung Yau Center of Southeast University, Southeast University, Nanjing 211189, China

^* Corresponding author
^* Corresponding author

Received: August 2020

Revised: October 2020

Early access: December 2020

Published: February 2021

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

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Abstract

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}) $.

Keywords:

Monge-Ampère equations,
Neumann problem,
degenerate equations,
global second order derivative estimates.

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

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