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 \(\alpha+\delta+\beta\le1, \beta>0\) 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.