–estimates for the Linearized Monge–ampère Equation

Let Ω ⊆ Rn be a strictly convex domain and let φ ∈ C2(Ω) be a convex function such that λ ≤ detD2φ ≤ Λ in Ω. The linearized Monge– Ampère equation is LΦu = trace(ΦD u) = f, where Φ = (detD2φ)(D2φ)−1 is the matrix of cofactors of D2φ. We prove that there exist p > 0 and C > 0 depending only on n, λ,Λ, and dist(Ω′,Ω) such that ‖Du‖Lp(Ω′) ≤ C(‖u‖L∞(Ω) + ‖f‖Ln(Ω)) for all solutions u ∈ C2(Ω) to the equation LΦu = f .