On Neumann problem for the degenerate Monge–Ampère type equations

Abstract

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