Existence and regularity of the solutions of some singular Monge–Ampère equations

Abstract

In this paper, we investigate the following singular Monge–Ampère equation(0.1){det⁡D2u=1(Hu)n+k+2u⁎kinΩ⊂⊂Rn,u=0,on∂Ω where k≥0, H<0 are constants and u⁎=x⋅∇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∈C∞(Ω)∩C(Ω¯) via regularization method. Using the technique in [10,12], we also obtain the optimal graph regularity of the solution of (0.1).