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 $ψσ_k(λ)^α$, where $α$ is a positive constant, $σ_k(λ)$ is the $k$-th elementary symmetric polynomial of the principal radii of curvature and $ψ$ is a preassigned positive smooth function defined on $\mathbb{S}^n$. We prove that under some assumptions of $ψ$, 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 $u^{1-p}( x ) σ_k \left( \nabla^2u+uI\right)=cψ(x)$ for $p>1$.

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