|
| 摘要: |
| 在Minkowksi型问题中, 一致$C^0$估计是一个重要且困难的问题. 本文考虑了广义Christoffel-Minkowski问题
\begin{equation}
\frac{\sigma_k(u_{ij}+u\delta_{ij})}{\sigma_l(u_{ij}+u\delta_{ij})} = u^{p-1}f(x), \notag \quad x \in \mathbb{S}^n,
\end{equation}
其中$0 \leq l < k \leq n$是整数, $ p-1 > 0$, $f$是一个正函数. 对于上述方程的的正凸偶解, 本文建立了解的加权梯度估计和一致$C^0$估计. 这是对Guan-Xia\cite{GX18}和Guan\cite{G2023}中结果的一般化. |
| 关键词: 加权梯度估计 凸解 Minkowski型问题 |
| DOI： |
| 分类号:O175.29 |
| 基金项目:国家自然科学基金项目（面上项目，重点项目，重大项目） |
|
| The UNIFORM $C^0$ ESTIMATE AND WEIGHTED ESTIMATE OF GENERALIZED CHRISTOFFEL-MINKOWSKI PROBLEMS |
|
zhangjinhu
|
| Abstract: |
| For the Minkowski type problems, the uniform $C^0$ estimate is an important and difficult issue. In this paper, we consider generalized Christoffel-Minkowski problems as follows
\begin{equation}
\frac{\sigma_k(u_{ij}+u\delta_{ij} )}{\sigma_l(u_{ij}+u\delta_{ij} )} = u^{p-1}f(x), \notag \quad x \in \mathbb{S}^n,
\end{equation}
where $0 \leq l < k \leq n$, $ p-1 > 0$ and $f$ is positive, and we establish the weighted gradient estimate and uniform $C^0$ estimate for the positive convex even solutions, which is a generalization of Guan-Xia \cite{GX18} and Guan \cite{G2023}. |
| Key words: Weighted gradient estimate convex solution Minkowski type problem |
