Interior gradient estimates for solutions to the linearized Monge–Ampère equation

Abstract

Let ϕ be a convex function on a convex domain Ω ⊂ R n , n ⩾ 1 . The corresponding linearized Monge–Ampère equation is trace ( Φ D 2 u ) = f , where Φ : = det D 2 ϕ ( D 2 ϕ ) − 1 is the matrix of cofactors of D 2 ϕ . We establish interior Hölder estimates for derivatives of solutions to such equation when the function f on the right-hand side belongs to L p ( Ω ) for some p > n . The function ϕ is assumed to be such that ϕ ∈ C ( Ω ¯ ) with ϕ = 0 on ∂ Ω and the Monge–Ampère measure det D 2 ϕ is given by a density g ∈ C ( Ω ) which is bounded away from zero and infinity.