2025/04/16 by Chen, Huan-Jie, Du, Shi-Zhong
#Analysis of PDEs (math.AP) #FOS: Mathematics
paper · doi:10.48550/arxiv.2504.11937
The affine maximal type hypersurface has been a core topic in Affine Geometry. When the hypersurface is presented as a regular graph of a convex function u, the statement that the graph is of affine maximal type is equivalent to the statement that u satisfies the fully nonlinear partial differential equation Dij(Uijw)=0, w≡[det D2u]-θ, θgt;0, ∀ x∈ℝN of fourth order. This equation can be regarded as a generalization of the N-dimensional Monge-Ampère equation det D2u=1, ∀ x∈ℝN of second order, since each solution of Monge-Ampère Equation satisfies affine maximal type equation automatically. In this paper, we will determine the symmetry groups of these two important fully nonlinear equations without asymptotic growth assumption. Our method develops the Lie's theory to fully nonlinear PDEs.