论文标题
在非均匀版本的燃烧问题上
On a non-homogeneous version of a problem of Firey
论文作者
论文摘要
我们研究Monge-ampère类型方程的唯一性\ begin {equination} \ label {eq-abstract} det（u_ {ij}+Δ_{ij} ij {ij} u）_ {i，j = 1} $ s^{n-1} $，其中$ u $是在球体上的支撑函数的限制，$ s^{n-1} $的凸体机构，该凸体中包含其内部中的原点和$ g：（0，\ infty）\ to（0，\ infty）$是一个连续的功能。该问题是由Firey（1974）引发的，在情况下，在$ g（θ）=θ^{ - 1} $的情况下，问$ u \ equiv 1 $是否是（*）的唯一解决方案。最近，Brendle，Choi和Daskalopoulos $ [9] $证明，如果$ g（θ）=θ^{ - p} $，$ p> -n-1 $，则$ u $必须是恒定的，特别是为Firey问题提供了完整的解决方案。我们的主要目标是为（*）获得更广泛的功能家庭$ g $的独特性（或几乎唯一）的结果。我们的方法与$ [9] $开发的技术大不相同。
We investigate the uniqueness for the Monge-Ampère type equation \begin{equation} \label{eq-abstract} det(u_{ij}+δ_{ij}u)_{i,j=1}^{n-1}=G(u),\ \ \ \ \ \ \ (*)\end{equation}on $S^{n-1}$, where $u$ is the restriction of the support function on the sphere $S^{n-1}$ of a convex body that contains the origin in its interior and $G:(0,\infty)\to(0,\infty)$ is a continuous function. The problem was initiated by Firey (1974) who, in the case $G(θ)=θ^{-1}$, asked if $u\equiv 1$ is the unique solution to (*). Recently, Brendle, Choi and Daskalopoulos $[9]$ proved that if $G(θ)=θ^{-p}$, $p>-n-1$, then $u$ has to be constant, providing in particular a complete solution to Firey's problem. Our primary goal is to obtain uniqueness (or nearly uniqueness) results for (*) for a broader family of functions $G$. Our approach is very different than the techniques developed in $[9]$.
