On Neumann problem for the degenerate Monge–Ampère type equations

Abstract

Abstract In this paper, we study the global C 1 , 1 $C^{1, 1}$ regularity for viscosity solution of the degenerate Monge–Ampère type equation det [ D 2 u − A ( x , D u ) ] = B ( x , u , D u ) $\det [D^{2}u-A(x, Du)]=B(x, u, Du)$ with the Neumann boundary value condition D ν u = φ ( x ) $D_{\nu }u=\varphi (x)$ , where the matrix A is under the regular condition and some structure conditions, and the right-hand term B is nonnegative.

Keywords

• Degenerate equations
• Monge–Ampère type equations
• Neumann problem
• Global second order derivative estimates
• Existence