A unified flow approach to smooth $L^p$ Christoffel-Minkowski problem for $p>1$

In this paper we study an anisotropic expanding flow of smooth, closed, uniformly convex hypersurfaces in ${\mathbb{R}}^{n+1}$ with speed $\psi {\sigma }_k(\lambda {)}^{\alpha }$, where $\alpha$ is a positive constant, ${\sigma }_k(\lambda )$ is the $k$-th elementary symmetric polynomial of the principal radii of curvature and $\psi$ is a preassigned positive smooth function defined on ${\mathbb{S}}^n$. We prove that under some assumptions of $\psi$, the solution to the flow after normalisation exists for all time and converges smoothly to a solution of the well-known $L^p$ Christoffel-Minkowski problem $u^{1-p}(x){\sigma }_k({\mathrm{\nabla }}^{2}u+uI)=c\psi (x)$ for $p>1$.