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

2024/04/01 by Weijun Zhang, Zhitao Zhang, Zhang, Weijun +1
Mathematics · #Advanced Differential Equations and Dynamical Systems #Analysis of PDEs (math.AP) #FOS: Mathematics #Functional Analysis (math.FA) #Nonlinear Differential Equations Analysis #Nonlinear Partial Differential Equations

paper · pdf · doi:10.48550/arxiv.2404.02925

openalex publication_date 2024/04/01 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

In this paper, we are concerned with the monotonic and symmetric properties of convex solutions to fully nonlinear elliptic systems. We mainly discuss Monge-Ampère type systems for instance, considering det(D²uⁱ)=fⁱ(x,\bf u,∇ uⁱ), 1≤ i≤ m, over bounded domains of various cases, including the bounded smooth simply connected domains and bounded tube shape domains in ℝⁿ. 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. The existence and uniqueness to an interesting example of such system is also discussed as an application of our results.

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