任意有界凸区域上退化Monge-Ampère方程的唯一性
Uniqueness for the Degenerate Monge-Ampère Equation on Arbitrary Bounded Convex Domains
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中文总结 AI 辅助
该研究针对任意有界凸区域上的退化Monge-Ampère方程，证明p>n时对应Dirichlet问题至多有一个凸Alexandrov解，p=n时同系数非零解互为正倍数，为该类方程的唯一性问题提供了理论依据。
中文摘要 AI 辅助
设n≥2，Ω是ℝⁿ中任意有界开凸集。作者证明，当p>n时，Dirichlet问题：在Ω内det D²u=(-u)^p，在∂Ω上u=0，在Ω内u>0，至多有一个凸Alexandrov解。证明基于Monge-Ampère能量的仿射行为，以及连接两个解的Legendre路径上L^{p+1}质量的幂凹性。在齐次指数p=n时，相同论证表明，具有相同系数的任意两个非零解互为正倍数。
英文摘要
Let $n\ge2$ and let $Ω\subset\mathbb R^n$ be an arbitrary bounded open convex set. The author prove that, for $p>n$, the Dirichlet problem \[ \det D^2u=(-u)^p\quad\text{in }Ω, \qquad u=0\quad\text{on }\partialΩ, \qquad u>0\quad\text{in }Ω\] has at most one convex Alexandrov solution. The proof is based on the affine behavior of the Monge--Ampère energy and on a power-concavity property of the $L^{p+1}$ mass along the Legendre path connecting two solutions. At the homogeneous exponent $p=n$, the same argument shows that any two nonzero solutions with the same coefficient are positive multiples of one another.
