Sharp Hessian integrability for the planar Monge--Ampère equation
Abstract
We determine the optimal local integrability of the Hessian for planar convex Alexandrov solutions satisfying λ dx≤μu≤λ−1 dx, 0<λ<1. The distributional Hessian satisfies the sharp endpoint estimate D2u∈ LlocP,∞, P=(1+λ)/(1−λ). This settles the weak estimate proposed by De Philippis and Figalli and identifies the larger optimal exponent P. We further construct strictly convex C1 solutions for which the distribution function of the Laplacian is comparable to T−P for all sufficiently large T. Hence both the exponent and the weak Lorentz endpoint are optimal: LP,∞ cannot be replaced by LP,b for any finite b>0. As applications, we obtain sharp estimates in Minty coordinates and optimal quantitative bi-Lipschitz restrictions for gradient maps, including an optimal exceptional-mass bound in quadratic-cost transport.
Related
- On a higher-integrability estimate for Laplacians of convex solutions to the Monge-Ampère equation
- Sharp conditions for fully nonlinear equations with (n−1)-Hessian matrix: Part I. Monge--Ampère case
- Sharp Hessian integrability for fully nonlinear elliptic supersolutions in low dimensions
- Triangles minimize the Monge-Ampère eigenvalue among planar bounded convex domains of a given area
- Quadratic growth rigidity for the complex Monge--Ampère equation
BibTeX
@misc{lu2026sharp,
title = {{Sharp Hessian integrability for the planar Monge--Ampère equation}},
author = {XiaoRui Lu and Jin Yan and Bao Yu and Yang Zhou},
howpublished = {arXiv},
year = {2026},
month = oct,
doi = {10.48550/arxiv.2610.21956},
eprint = {2610.21956},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2610.21956},
}
RIS
TY - UNPB
TI - Sharp Hessian integrability for the planar Monge--Ampère equation
AU - XiaoRui Lu
AU - Jin Yan
AU - Bao Yu
AU - Yang Zhou
PY - 2026
DA - 2026/10/15
PB - arXiv
DO - 10.48550/arxiv.2610.21956
UR - https://arxiv.org/abs/2610.21956
AB - We determine the optimal local integrability of the Hessian for planar convex Alexandrov solutions satisfying λ dx≤μ_u≤λ^−1 dx, 0<λ<1. The distributional Hessian satisfies the sharp endpoint estimate D^2u∈ L_loc^P,∞, P=(1+λ)/(1−λ). This settles the weak estimate proposed by De Philippis and Figalli and identifies the larger optimal exponent P. We further construct strictly convex C^1 solutions for which the distribution function of the Laplacian is comparable to T^−P for all sufficiently large T. Hence both the exponent and the weak Lorentz endpoint are optimal: L^P,∞ cannot be replaced by L^P,b for any finite b>0. As applications, we obtain sharp estimates in Minty coordinates and optimal quantitative bi-Lipschitz restrictions for gradient maps, including an optimal exceptional-mass bound in quadratic-cost transport.
ER -