2014/10/07 by Mostafa Fazly, Fazly, Mostafa
Computer Science · Mathematics · #Advanced Mathematical Modeling in Engineering #Analysis of PDEs (math.AP) #FOS: Mathematics #Geometric Analysis and Curvature Flows #Nonlinear Partial Differential Equations #math.AP
paper · pdf · doi:10.48550/arxiv.1410.1831
To appear in Proc. Amer. Math. Soc. 15 pages. Comments are welcome. See http://www.math.ualberta.ca/~fazly/research.html for updates
arxiv created 2014/10/07 · openalex publication_date 2014/10/07 · arxiv updated 2014/10/08 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We study stable solutions of the following nonlinear system -Δu = H(u) in Ω where u:\mathbb Rn→ \mathbb Rm, H:\mathbb Rm→ \mathbb Rm and Ω is a domain in \mathbb Rn. We introduce the novel notion of symmetric systems. The above system is said to be symmetric if the matrix of gradient of all components of H is symmetric. It seems that this concept is crucial to prove Liouville theorems, when Ω=\mathbb Rn, and regularity results, when Ω=B1, for stable solutions of the above system for a general nonlinearity H ∈ C1(\mathbb R m). Moreover, we provide an improvement for a linear Liouville theorem given in [20] that is a key tool to establish De Giorgi type results in lower dimensions for elliptic equations and systems.