On Boundary Hölder Gradient Estimates for Solutions to the Linearized Monge-ampère Equations

In this paper, we establish boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations with Lp (n < p ≤ ∞) right hand side and C1,γ boundary values under natural assumptions on the domain, boundary data and the MongeAmpère measure. These estimates extend our previous boundary regularity results for solutions to the linearized Monge-Ampère equations with bounded right hand side and C1,1 boundary data. 1. Statement of the main results In this paper, we establish boundary Hölder gradient estimates for solutions to the linearized Monge-Ampère equations with Lp (n < p ≤ ∞) right hand side and C1,γ boundary values under natural assumptions on the domain, boundary data and the Monge-Ampère measure. Before stating these estimates, we introduce the following assumptions on the domain Ω and function φ. Let Ω ⊂ Rn be a bounded convex set with (1.1) Bρ(ρen) ⊂ Ω ⊂ {xn ≥ 0} ∩ B 1 ρ , for some small ρ > 0. Assume that (1.2) Ω contains an interior ball of radius ρ tangent to ∂Ω at each point on ∂Ω ∩ Bρ. Let φ : Ω → R, φ ∈ C0,1(Ω) ∩ C2(Ω) be a convex function satisfying (1.3) 0 < λ ≤ det Dφ ≤ Λ in Ω. Throughout, we denote by Φ = (Φi j) the matrix of cofactors of the Hessian matrix D2φ, i.e., Φ = (det Dφ)(Dφ). Mathematics Subject Classification (2010): 35J70, 35B65, 35B45, 35J96.