Saved in:
| Main Authors: | , |
|---|---|
| Format: | Preprint |
| Published: |
2025
|
| Subjects: | |
| Online Access: | https://arxiv.org/abs/2504.02661 |
| Tags: |
Add Tag
No Tags, Be the first to tag this record!
|
| _version_ | 1866915225980108800 |
|---|---|
| author | Chen, Huan-Jie Du, Shi-Zhong |
| author_facet | Chen, Huan-Jie Du, Shi-Zhong |
| contents | In Convex Geometry, a core topic is the $L_p$-Minkowski problem
\begin{equation}\label{e0.1}
\det(\nabla^2h+hI)=fh^{p-1}, \ \ \forall X\in{\mathbb{S}}^n, \ \ \forall p\in \mathbb{R}
\end{equation} of Monge-Ampère type. By the transformation $u(x)=h(X)\sqrt{1+|x|^2}$ and semi-spherical projection, equation \eqref{e0.1} can be reformulated by the Monge-Ampère type equation
\begin{equation}\label{e0.2}
\det D^2u=(1+|x|^2)^{-\frac{p+n+1}{2}}u^{p-1}, \ \ \forall x\in{\mathbb{R}}^n, \ \ \forall p\in \mathbb{R}
\end{equation} on the Euclidean space. In this paper, we will firstly determine the symmetric groups of $n$-dimensional fully nonlinear equation \eqref{e0.2} without asymptotic growth assumption. After proving several key resolution lemmas, we thus completely classify the symmetric groups of the $L_p$-Minkowski problem. Our method develops the Lie theory to fully nonlinear PDEs in Convex Geometry. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_02661 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Complete Classification of the Symmetry Group of $L_p$-Minkowski Problem on the Sphere Chen, Huan-Jie Du, Shi-Zhong Analysis of PDEs Differential Geometry In Convex Geometry, a core topic is the $L_p$-Minkowski problem \begin{equation}\label{e0.1} \det(\nabla^2h+hI)=fh^{p-1}, \ \ \forall X\in{\mathbb{S}}^n, \ \ \forall p\in \mathbb{R} \end{equation} of Monge-Ampère type. By the transformation $u(x)=h(X)\sqrt{1+|x|^2}$ and semi-spherical projection, equation \eqref{e0.1} can be reformulated by the Monge-Ampère type equation \begin{equation}\label{e0.2} \det D^2u=(1+|x|^2)^{-\frac{p+n+1}{2}}u^{p-1}, \ \ \forall x\in{\mathbb{R}}^n, \ \ \forall p\in \mathbb{R} \end{equation} on the Euclidean space. In this paper, we will firstly determine the symmetric groups of $n$-dimensional fully nonlinear equation \eqref{e0.2} without asymptotic growth assumption. After proving several key resolution lemmas, we thus completely classify the symmetric groups of the $L_p$-Minkowski problem. Our method develops the Lie theory to fully nonlinear PDEs in Convex Geometry. |
| title | Complete Classification of the Symmetry Group of $L_p$-Minkowski Problem on the Sphere |
| topic | Analysis of PDEs Differential Geometry |
| url | https://arxiv.org/abs/2504.02661 |