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 D2 u=M|u|pν with zero boundary values on bounded convex domains in ℝn (n ≥ 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 ∫Ωdist (·,∂Ω) dν<∞. 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=dist (·,∂Ω)−α dLn, 0≤α<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≤ 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.
Related
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- Global regularity for a class of degenerate/singular Monge-Ampère equations
BibTeX
@misc{gu2026uniqueness,
title = {{Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures}},
author = {Chong Gu},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.27341},
eprint = {2609.27341},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.27341},
}
RIS
TY - UNPB
TI - Uniqueness and sharp boundary estimates for degenerate Monge-Ampère equations with singular measures
AU - Chong Gu
PY - 2026
DA - 2026/09/23
PB - arXiv
DO - 10.48550/arxiv.2609.27341
UR - https://arxiv.org/abs/2609.27341
AB - 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 ℝ^n (n ≥ 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 ∫_Ωdist (·,∂Ω) dν<∞. 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=dist (·,∂Ω)^−α dL^n, 0≤α<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≤ 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.
ER -