In this paper, we establish various maximum principles and develop the method of moving planes for equations involving the uniformly elliptic nonlocal Bellman operator. As a consequence, we derive multiple applications of these maximum principles and the moving planes method. For instance, we prove symmetry, monotonicity and uniqueness results and asymptotic properties for solutions to various equations involving the uniformly elliptic nonlocal Bellman operator in bounded domains, unbounded domains, epigraph or \({\mathbb {R}}^{n}\). In particular, the uniformly elliptic nonlocal Monge–Ampère operator introduced by Caffarelli and Charro (Ann PDE 1:4, 2015) is a typical example of the uniformly elliptic nonlocal Bellman operator.

在本文中，我们建立了各种最大原理，并为涉及均匀椭圆形非局部Bellman算子的方程式开发了移动平面的方法。结果，我们得出了这些最大原理和移动平面方法的多种应用。例如，我们证明了在有界域，无界域，题词或\（{\ mathbb {R}} ^ {n} \）中涉及均匀椭圆非局部Bellman算子的各种方程的解的对称性，单调性和唯一性结果以及渐近性质。特别是，由Caffarelli和Charro（Ann PDE 1：4，2015）引入的均匀椭圆非局部Monge–Ampère算子是均匀椭圆非局部Bellman算子的典型示例。