Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures

Abstract

We study the uniqueness and boundary behavior of nonzero convex Aleksandrov solutions to $\det D^{2}u=M|u{|}^p\nu$ with zero boundary values on bounded convex domains in ${\mathbb{R}}^n(n\ge 2)$. For $0<p<n$, we prove the uniqueness of nonzero convex solutions in the finite-energy class when $\nu$ is a locally finite Borel measure with positive mass and ${\int }_{\Omega }\text{dist }(\cdot ,\partial \Omega )d\nu <\infty$. 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 $\nu =\text{dist }(\cdot ,\partial \Omega {)}^{-\alpha }d{L}^n$, $0\le \alpha <2$, we prove global Lipschitz continuity when $p-\alpha >n-2$ and obtain sharp upper and lower estimates on domains with a flat boundary part when $n(\alpha -1)-2<p-\alpha \le n-2$. When $\alpha =0$ 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.

32 pages. ChatGPT 6 Astra is used calculations and language improvements. The key ideas and proofs are from the author. The author checked the calculations and takes full responsibility for their correctness