Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures
Year: 2026
Type: preprint
Abstract: 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 $0n$, 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
Source: arXiv (Cornell University)
Author: Gu, Chong
Language: English
Subfield: Applied Mathematics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
Funder: National Science Foundation
OpenAlex ID: W7214148428