A Curvature Flow Approach to \(L^p\) Christoffel-Minkowski Problem for \(1

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