Existence and regularity of the solutions of some singular Monge-Ampere equations

In this paper, we investigate the following singular Monge-Ampere equation {det D(2)u = 1/(Hu)(n+k+2)u*(k) in Omega subset of subset of R-n, (0.1) u = 0, on partial derivative Omega where k >= 0, H < 0 are constants and u* = x center dot del u(x) - u(x) is the Legendre transformation of u. Equation (0.1) is related to proper affine hyperspheres. We will show the existence of solutions of (0.1) u epsilon C-infinity(Omega) boolean AND C(<(Omega)over bar>) via regularization method. Using the technique in [10,12], we also obtain the optimal graph regularity of the solution of (0,1). (C) 2019 Elsevier Inc. All rights reserved.