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

Abstract

In this paper, we are concerned with the monotonic and symmetric properties of convex solutions Monge-Ampère systems for instance, considering \begin{equation*} \det(D^2u^i)=f^i(x,{\bf u},\nabla u^i), \ 1\leq i\leq m, \end{equation*} over unbounded domains of various cases, including the whole spaces ${\mathbb{R}}^n$, the half spaces ${\mathbb{R}}_{+}^n$ and the unbounded tube shape domains in ${\mathbb{R}}^n$. 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.

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