论文标题
一类逆弯曲流和$ l^p $双christoffel-minkowski问题
A class of inverse curvature flows and $L^p$ dual Christoffel-Minkowski problem
论文作者
论文摘要
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 $ψu^αρ^δf^{-β}$, where $ψ$ is a smooth positive function on unit sphere, $u$ is the support function of the hypersurface, $ρ$ is the radial function, $ f $是一种平滑，对称性，同质的第一度，是凸锥上高表面的主要曲率的正功能。当$ψ= 1 $时，我们证明该流量存在一直存在，如果$α+δ+δ+δ+β\ le1，β> 0 $和$α\ le0 $，而如果$α+α+α+Δ+β> 1，α，α，δ\ le0 $，则在有限的时间和我们假定hystruce the Primate hystruce hyperface中浮出水面。在这两种情况下，正确重新缩放的流都汇聚到以原点为中心的球体。特别是，可以通过放置$α=δ= 0 $来恢复Gerhardt \ cite {gc，gc3}和urbas \ cite {uj2}的结果。我们以前的作品\ cite {dl，dl2}可以通过放置$δ= 0 $来恢复。通过这些流的融合，我们可以提供新的独特定理证明解决方案，以解决$ l^p $ -minkowski问题，以及$ l^p $ -Christoffel-Minkowski问题，并具有常数处方数据。同样，我们提出了$ l^p $双重克里斯托佛尔·米科夫斯基问题，并证明了解决方案的独特定理，用于$ l^p $ dual minkowski问题和$ l^p $ DUAL CHRISTOFFEL-MINKOWSKI问题，并带有常规规定的数据。最后，我们专注于一类各向异性流的长期存在和收敛性（即，对于一般函数$ψ$）。最终结果不仅提供了许多先前已知的解决方案的新证明，用于$ l^p $ Dual Minkowski问题，$ L^p $ -Christoffel-Minkowski问题等。
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 $ψu^αρ^δf^{-β}$, where $ψ$ is a smooth positive function on unit sphere, $u$ is the support function of the hypersurface, $ρ$ 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 $ψ=1$, we prove that the flow exists for all time and converges to infinity if $α+δ+β\le1, β>0$ and $α\le0$, while in case $α+δ+β>1,α,δ\le0$, 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 $α=δ=0$. Our previous works \cite{DL,DL2} can be recovered by putting $δ=0$. 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 $ψ$). 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.
