论文标题
Monge-ampère方程系统的独特性
Uniqueness for a system of Monge-Ampère equations
论文作者
论文摘要
在本说明中，我们证明了一个唯一性结果，直至一个正乘式常数，对于对Monge-ampère方程系统的非平凡凸解决方案\ begin \ begin {equation*} \ left \ left \ {\ okent {alignEdat} {alignEdat} {2} {2} {2} {2} {2} {2} {2} {2} {2} \ det d^2 d^2 u〜〜〜〜〜〜〜〜g | v |^p〜p〜poug | d^2 v〜＆=μ| u |^{n^2/p}〜&& \ text {in} 〜Ω，\\\ u = v＆= v＆= 0〜 && \ text {on}〜\partialΩ \ end {equation*}上有界，平滑且均匀的凸域$ω\ subset r^n $，提供$ p $接近$ n \ geq 2 $。当$ p = n $时，我们证明了通用界凸域$ω\ subset r^n $的唯一性。
In this note, we prove a uniqueness result, up to a positive multiplicative constant, for nontrivial convex solutions to a system of Monge-Ampère equations \begin{equation*} \left\{ \begin{alignedat}{2} \det D^2 u~& = γ|v|^p~&&\text{in} ~ Ω, \\\ \det D^2 v~& = μ|u|^{n^2/p}~&&\text{in} ~ Ω, \\\ u=v &= 0~&&\text{on}~ \partialΩ\end{alignedat} \right. \end{equation*} on bounded, smooth and uniformly convex domains $Ω\subset R^n$ provided that $p$ is close to $n\geq 2$. When $p=n$, we show that the uniqueness holds for general bounded convex domains $Ω\subset R^n$.
