Symmetry of Convex Solutions to Fully Nonlinear Elliptic Systems: Unbounded Domains

In this paper, we are concerned with the monotonic and symmetric properties of convex solutions Monge-Ampère systems for instance, considering (D^2u^i)=f^i(x, u,∇ u^i), 1≤ i≤ m, over unbounded domains of various cases, including the whole spaces ℝ^n, the half spaces ℝ^n_+ and the unbounded tube shape domains in ℝ^n. We obtain monotonic and symmetric properties of the solutions to the problem with respect to the geometry of domains and the monotonic and symmetric properties of right-hand side terms. The proof is based on carefully using the moving plane method together with various maximum principles and Hopf's lemmas.