标题： 新的混合面积积分和非凸区域的Poincaré型不等式 显示英文标题

标题： New quermassintegral and Poincaré type inequalities for non-convex domains

摘要： 在本文的第一部分中，我们研究了欧几里得空间中的以下非齐次、局部约束的逆曲率流 $\mathbb{R}^{n+1}$，\begin{align*} \dot{x}=\left(\frac{1}{\frac{E_k(\hat{\kappa})}{E_{k-1}(\hat{\kappa})}-\alpha }-\langle x,\nu\rangle\right)\nu, \quad k=2,3,\ldots,n-1. \end{align*}假设初始超曲面$\mathcal{M}_0 \subset \mathbb{R}^{n+1}$是星形的，并且其平移后的主曲率 $\hat{\kappa}=\kappa+\alpha(1,\ldots,1)$位于凸集\begin{align*} \Gamma_{\alpha,k}:=\Gamma_{k-1}\cap \{\lambda\in \mathbb{R}^n:\, E_k(\lambda)-\alpha E_{k-1}(\lambda)>0\}, \end{align*}中，我们证明该流存在全局光滑解，并且会光滑地收敛到一个球面。 作为推论，我们得到了非凸区域的一组新的亚历山大-芬赫尔型不等式。 在第二部分中，我们推导出一种对于$k$-凸超曲面的庞加莱型不等式，这补充了著名的 海因茨-卡彻不等式的一个更一般的版本。 显示英文摘要

摘要： In the first part of this paper, we study the following non-homogeneous, locally constrained inverse curvature flow in Euclidean space $\mathbb{R}^{n+1}$, \begin{align*} \dot{x}=\left(\frac{1}{\frac{E_k(\hat{\kappa})}{E_{k-1}(\hat{\kappa})}-\alpha }-\langle x,\nu\rangle\right)\nu, \quad k=2,3,\ldots,n-1. \end{align*} Assuming that the initial hypersurface $\mathcal{M}_0 \subset \mathbb{R}^{n+1}$ is star-shaped and its shifted principal curvatures $\hat{\kappa}=\kappa+\alpha(1,\ldots,1)$ lie in the convex set \begin{align*} \Gamma_{\alpha,k}:=\Gamma_{k-1}\cap \{\lambda\in \mathbb{R}^n:\, E_k(\lambda)-\alpha E_{k-1}(\lambda)>0\}, \end{align*} we show that the flow admits a smooth solution that exists for all positive times, and it converges smoothly to a round sphere. As a corollary, we obtain a new set of Alexandrov-Fenchel-type inequalities for non-convex domains. In the second part, we derive a Poincar\'{e} type inequality for $k$-convex hypersurfaces which complements a more general version of the well-known Heintze-Karcher inequality.