带外力项的曲率半径热流方程

On a Heat Flow Equation for Curvature Radii with Forced Term

摘要: 在本文中，我们证明下述热流方程在${\mathbb{S}}^n\times \left[0,T\right)$ 上存在光滑凸解的短时间存在性$\begin{array}{c}\{\begin{array}{l}{X}_t=\left({\left(\frac{{\sigma }_n}{{\sigma }_k}\right)}^{\frac{1}{n-k}}\left(\lambda \left(h_{ij}\right)\right)-u^q{\varrho }^{p}f\left(p\right)\right)\cdot \nu ,\left(z,t\right)\in {\mathbb{S}}^n\times \left[0,T\right);\hfill \\ X\left(z,0\right)=\phi \left(z\right)\cdot \nu ,z\in {\mathbb{S}}^n\hfill \end{array}\end{array}$ 该研究源于一些带外力项的热流问题以及Minkowski问题的热流方法。文中还给出了曲率估计和凸性估计，这是由$h_{ij}$ 的最大特征值和最小特征值的一些微分不等式推导而来的。

Abstract: In the present paper, we prove a short-time existence of smooth convex solutions to the following heat flow equation on ${\mathbb{S}}^n\times \left[0,T\right)$ $\begin{array}{c}\{\begin{array}{l}{X}_t=\left({\left(\frac{{\sigma }_n}{{\sigma }_k}\right)}^{\frac{1}{n-k}}\left(\lambda \left(h_{ij}\right)\right)-u^q{\varrho }^{p}f\left(p\right)\right)\cdot \nu ,\left(z,t\right)\in {\mathbb{S}}^n\times \left[0,T\right);\hfill \\ X\left(z,0\right)=\phi \left(z\right)\cdot \nu ,z\in {\mathbb{S}}^n\hfill \end{array}\end{array}$ which was motivated by some heat flow problems with forced term and heat flow approach to Minkowski type problem. The curvature estimates and convexity estimates follows from some differential inequalities of largest eigenvalue and smallest eigenvalue of $h_{ij}$ respectively.