A class of inverse curvature flows and $L^p$ dual Christoffel-Minkowski problem

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 $\alpha +\delta +\beta \le 1,\beta >0$ and $\alpha \le 0$, while in case $\alpha +\delta +\beta >1,\alpha ,\delta \le 0$, 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 and Urbas can be recovered by putting $\alpha =\delta =0$. Our previous works 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.