We investigate the Monge-Ampere equation subject to zero boundary value and with a positive right-hand side unction 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 nonlinear dependence between the Holder seminorms of the solution and of the right-side function, which confirms the results of Figalli, Jhaveri and Mooney (J. Func. Anal. 2016). Our technique is in part inspired by Jian and Wang (SIAM J. Math. Anal. 2007) which includes using a sequence of so-called sections.

中文翻译：

---

Monge-Ampère 方程在连续性模数方面的内部估计

我们研究了受零边界值约束的 Monge-Ampere 方程，并且假设右手边的正函数是连续的或基本上有界的。根据连续性模数获得解的一阶和二阶导数的内部估计。我们解释了估计如何依赖于各种数量，但让它们独立于解决方案的凸性模量。我们的主要定理有许多有用的推论。其中之一是解的 Holder 半范数与右侧函数之间的非线性相关性，这证实了 Figalli、Jhaveri 和 Mooney (J. Func. Anal. 2016) 的结果。我们的技术部分受到Jian 和Wang（SIAM J. Math. Anal. 2007）的启发，其中包括使用一系列所谓的部分。