Interior estimates for Monge-Ampere equation in terms of modulus of continuity

摘要

We investigate the Monge-Ampere equation subject to zero boundary value and with a positive right-hand side function assumed to be continuous or essentially bounded. Interior estimates of the solution's first and second derivatives are obtained in terms of moduli of continuity. We explicate how the estimates depend on various quantities but have them independent of the solution's modulus of convexity. Our main theorem has many useful consequences. One of them is the nonlineardependence between the Holder seminorms of the solution and of the right-side function, which confirms the results of Figalli, Jhaveri & Mooney in [7]. Our technique is in part inspired by Jian & Wang in [11] which includes using a sequence of so-called sections. (C) 2020 The Authors. Published by Elsevier Inc.