Complete Classification of the Symmetry Groups of Monge-Ampère Equation and Affine Maximal type Equation
Huan-Jie Chen, Shi-Zhong Du
TL;DR
This work delivers a complete Lie-symmetry classification for two central nonlinear PDEs in affine geometry: the Monge-Ampère equation $\det D^2 u=1$ and the affine maximal type equation $u^{ij} D_{ij} w=0$ with $w=[\det D^2 u]^{-\theta}$. By developing a fourth-order prolongation framework and applying Lie group analysis, the authors derive explicit infinitesimal generators and global group actions, providing a full description of symmetries without any growth assumptions. A key result is that the Monge-Ampère equation admits a rich, explicit symmetry algebra, while the affine maximal type equation generally shares the same symmetries except at the critical $\theta=\frac{N+1}{N+2}$, where an enhanced $SO(N+1)$-type symmetry emerges. These classifications enable systematic symmetry reductions and contribute to understanding Bernstein-type phenomena through a complete symmetry lens in higher dimensions.
Abstract
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 $$ D_{ij}(U^{ij}w)=0, \ \ w\equiv[\det D^2u]^{-θ}, \ \ θ>0, \ \ \forall x\in{\mathbb{R}}^N $$ of fourth order. This equation can be regarded as a generalization of the $N$-dimensional Monge-Ampère equation $$ \det D^2u=1, \ \ \forall x\in{\mathbb{R}}^N $$ 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.