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

Let ϕ   be a convex function on a convex domain Ω⊂RnΩ⊂Rn, n⩾1n⩾1. The corresponding linearized Monge–Ampère equation istrace(ΦD2u)=f,trace(ΦD2u)=f, where Φ:=detD2ϕ(D2ϕ)−1 is the matrix of cofactors of D2ϕD2ϕ. We establish interior Hölder estimates for derivatives of solutions to such equation when the function f   on the right-hand side belongs to Lp(Ω)Lp(Ω) for some p>np>n. The function ϕ   is assumed to be such that ϕ∈C(Ω¯) with ϕ=0ϕ=0 on ∂Ω   and the Monge–Ampère measure detD2ϕ is given by a density g∈C(Ω)g∈C(Ω) which is bounded away from zero and infinity.