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 ⁡ D 2 u = 1 (H u) n + k + 2 u ⁎ k in Ω ⊂ ⊂ R n , 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). [ABSTRACT FROM AUTHOR]