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

Abstract

Let φ be a convex function on a convex domain Sigma;CRn, n≥1. The corresponding linearized Monge-Ampère equation is. trace(φD2u)=f, where φ:=detD2φ(D2φ)-1 is the matrix of cofactors of D2φ. 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) for some p>n. The function φ is assumed to be such that φbSigma;C(Ω) with φ=0 on ∂. Ω and the Monge-Ampère measure detD2φ is given by a density gSigma;C(Sigma;) which is bounded away from zero and infinity. © 2011 Elsevier Inc.