具有奇异测度的退化Monge-Ampère方程的唯一性与尖锐边界估计
Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures
- Indiana University（印第安纳大学）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
研究退化Monge-Ampère方程带奇异测度的非零凸解，证明了特定条件下的唯一性、构造了多解例子，并给出了尖锐边界估计及有限能量条件。
AI中文摘要：
我们研究在$\mathbb{R}^n (n \geq 2)$中有界凸域上具有零边界值的非零凸Aleksandrov解$\det D^2 u=M|u|^pν$的唯一性和边界行为。对于$0<p<n$，当$ν$是具有正质量的局部有限Borel测度且$\int_Ω\text{dist }(\cdot,\partialΩ)\\,dν<\infty$时，我们证明了有限能量类中非零凸解的唯一性。对于$p>n$，我们在单位球上构造了一个显式的两壳层测度，使得该问题至少有三个径向解，这些解全局Lipschitz且具有有限能量。在$ν=\text{dist }(\cdot,\partialΩ)^{-α}\\,dL^n$，$0\leqα<2$的情形下，当$p-α>n-2$时我们证明了全局Lipschitz连续性，并在$n(α-1)-2<p-α\leq n-2$时获得了具有平坦边界部分的域上的尖锐上下估计。当$α=0$且$p=n-2$时，我们的log-Lipschitz下估计与已知上估计具有相同的指数。这回答了Le提出的问题（Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains. Ann. Fac. Sci. Toulouse Math. (6) 35 (2026)）。我们还给出了在每个有界凸域上有限Monge-Ampère能量的充分条件，证明了当边界包含平坦部分时该条件的必要性，并应用它证明了在所有非零凸解中Monge-Ampère特征值的唯一性。
英文摘要：
We study the uniqueness and boundary behavior of nonzero convex Aleksandrov solutions to $\det D^2 u=M|u|^pν$ with zero boundary values on bounded convex domains in $\mathbb{R}^n (n \geq 2)$. For $0<p<n$, we prove the uniqueness of nonzero convex solutions in the finite-energy class when $ν$ is a locally finite Borel measure with positive mass and $\int_Ω\text{dist }(\cdot,\partialΩ)\,dν<\infty$. For $p>n$, we construct an explicit two-shell measure on the unit ball for which the problem has at least three radial solutions that are globally Lipschitz and have finite energy. In the case of $ν=\text{dist }(\cdot,\partialΩ)^{-α}\,dL^n$, $0\leqα<2$, we prove global Lipschitz continuity when $p-α>n-2$ and obtain sharp upper and lower estimates on domains with a flat boundary part when $n(α-1)-2<p-α\leq n-2$. When $α=0$ and $p=n-2$, our log-Lipschitz lower estimate has the same exponent as the known upper estimate. This answers the question raised by Le (Global Lipschitz and Sobolev estimates for the Monge-Ampère eigenfunctions of general bounded convex domains. Ann. Fac. Sci. Toulouse Math. (6) 35 (2026)). We also give a sufficient condition for finite Monge-Ampère energy on every bounded convex domain, prove its necessity when the boundary contains a flat part, and apply it to prove the uniqueness of the Monge-Ampère eigenvalue among all nonzero convex solutions.
