Uniqueness and sharp boundary estimates for degenerate Monge-Amp\`ere 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 $\mathbb{R}^n (n \geq 2)$. 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 $\nu=\text{dist }(\cdot,\partial\Omega)^{-\alpha}\,dL^n$, $0\leq\alpha<2$, 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 $n(\alpha-1)-2<p-\alpha\leq n-2$. 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-Amp\`ere eigenfunctions of general bounded convex domains. Ann. Fac. Sci. Toulouse Math. (6) 35 (2026)). We also give a sufficient condition for finite Monge-Amp\`ere 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-Amp\`ere eigenvalue among all nonzero convex solutions.