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

Weijun Zhang; Zhitao Zhang

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arXiv · 2404.01185

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

Weijun Zhang·Zhitao Zhang

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Abstract

In this paper, we are concerned with the monotonic and symmetric properties of convex solutions Monge-Amp\`ere 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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Weijun Zhang, Zhitao Zhang. 2024-04-01. Symmetry of Convex Solutions to Fully Nonlinear Elliptic Systems: Unbounded Domains. https://arxiv.org/abs/2404.01185

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