arXiv2022
In this paper, we consider a large class of expanding flows of closed, smooth, star-shaped hypersurface in Euclidean space $\mathbb{R}^{n+1}$ with speed $\psi u^\alpha\rho^\delta f^{-\beta}$, where $\psi$ is a smooth positive function on unit sphere, $u$ is the support function of the hypersurface, $\rho$ is the radial function, $f$ is a smooth, symmetric, homogenous of degree one, positive function of the principal curvatures of the hypersurface on a convex cone. When $\psi=1$, we prove that the flow exists for all time and converges to infinity if $\alpha+\delta+\beta\le1, \beta>0$ and $\alpha\le0$, while in case $\alpha+\delta+\beta>1,\alpha,\delta\le0$, the flow blows up in finite time, and where we assume the initial hypersurface to be strictly convex. In both cases the properly rescaled flows converge to a sphere centered the origin. In particular, the results of Gerhardt \cite{GC,GC3} and Urbas \cite{UJ2} can be recovered by putting $\alpha=\delta=0$. Our previous works \cite{DL,DL2} can be recovered by putting $\delta=0$. By the convergence of these flows, we can give a new proof of uniqueness theorems for solutions to $L^p$-Minkowski problem and $L^p$-Christoffel-Minkowski problem with constant prescribed data. Similarly, we pose the $L^p$ dual Christoffel-Minkowski problem and prove a uniqueness theorem for solutions to $L^p$ dual Minkowski problem and $L^p$ dual Christoffel-Minkowski problem with constant prescribed data. At last, we focus on the longtime existence and convergence of a class of anisotropic flows (i.e. for general function $\psi$). The final result not only gives a new proof of many previously known solutions to $L^p$ dual Minkowski problem, $L^p$-Christoffel-Minkowski problem, etc. by such anisotropic flows, but also provides solutions to $L^p$ dual Christoffel-Minkowski problem with some conditions.