Maximum principles and the method of moving planes for the uniformly elliptic nonlocal Bellman operator and applications

Abstract

In this paper, we establish various maximum principles and develop the method of moving planes for equations involving the uniformly elliptic nonlocal Monge-Ampère operator introduced by Caffarelli and Charro in [14]. 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 Monge-Ampère operator in bounded domains, unbounded domains, epigraph or R n .